Angles of Triangles. Objectives Find angle measures in triangles.
Unit 6 Properties of Angles and Triangles · Unit 6 Properties of Angles and Triangles 18 A Regular...
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Unit 6 Properties of Angles and Triangles 1 Section 2.1 Exploring Parallel Lines (I) What are parallel lines? Example 1: When land is developed to form a network of roads within a community roads are often running parallel to each other. Example 2: When a wall is constructed for a home, studs run parallel to each other. Parallel Lines •are lines that ____________________________________ How could we determine if the studs are parallel to each other? Goals: Defining Parallel and Transversal Lines Identifying relationships among the measures of angles formed by intersecting lines.
Transcript of Unit 6 Properties of Angles and Triangles · Unit 6 Properties of Angles and Triangles 18 A Regular...
Unit 6 Properties of Angles and Triangles 1
Section 2.1 Exploring Parallel Lines
(I) What are parallel lines?
Example 1: When land is developed to form a network of roads within a community
roads are often running parallel to each other.
Example 2: When a wall is constructed for a home, studs run parallel to each other.
Parallel Lines •are lines that ____________________________________
How could we determine if the studs are parallel to each other?
Goals:
Defining Parallel and Transversal Lines
Identifying relationships among the measures of angles formed by
intersecting lines.
Unit 6 Properties of Angles and Triangles 2
(II) What are transversal lines?
Example: Windows panes often have decorative grills installed.
Transversal Lines •are lines that ____________________________________
(III) Angles formed by intersecting lines
Based on the diagram, state
the angles defined below.
Interior Angles •Any angle formed by a transversal and two parallel lines
that lie inside the parallel lines.
_____________________________________________
Exterior Angles •Any angle formed by a transversal and two parallel lines
that lie outside the parallel lines.
_____________________________________________
Transversal Line
Unit 6 Properties of Angles and Triangles 3
(III) Angles formed by intersecting lines
Based on the diagram, state
the angles defined below.
Corresponding Angles •One interior and one exterior angle that are
located on the same side of the transversal line
AND the parallel line.
_________________________________________
http://www.mathwarehouse.com/geometry/angle/interactive-transveral-angles.php
Note: When a transversal intersects two parallel lines, the corresponding
angles are ____________
(IV) Remembering Angle Properties
Supplementary Angles
•two angles that add to 180°.
Example: Determine the value of x for:
Vertically Opposite Angles
•angles formed by the intersection of two lines that are opposite each other.
Example: Determine the value of x for:
Transversal Line
125° x
60°
x
Unit 6 Properties of Angles and Triangles 4
(V) Converse
•A statement formed by switching the premise and conclusion
Example: State the converse of the statement below.
Statement: When a transversal intersects two parallel lines,
the corresponding angles are congruent.
Converse:
Example: Determine if lines m and n are parallel.
(1) (2)
Example: Determine the value of x if lines m and n are parallel.
m
n
30°
160°
m
n
92°
88°
m
n
2x + 10
160°
Questions: P.72 #2, #5
Unit 6 Properties of Angles and Triangles 5
Section 2.2 Angles Formed by Parallel Lines
(I) Review of Last Day
●Corresponding Angles formed by a
transversal intersecting parallel lines
are congruent.
Identify the angles.
___________
___________
___________
___________
●Supplementary Angles two angles that add to 180°.
●Vertically Opposite Angles angles formed by the intersection of two lines
that are opposite each other.
1 2
1
2
Transversal Line
Goals:
Prove properties of angles formed by parallel lines and a
transversal, and use these properties to solve problems.
Unit 6 Properties of Angles and Triangles 6
(II) Alternate Interior, Alternate Exterior and
Interior Angles on same side of transversal.
●Alternate Interior Angles
Two non-adjacent interior angles on
opposite sides of a transversal.
Identify the angles.
___________ ___________
●Alternate Exterior Angles
Two exterior angles formed between two lines and a transversal, on opposite
sides of a transversal.
Identify the angles.
___________ ___________
●Same Side Interior Angles
Angles formed between two lines and a transversal that lie in the interior and the same
side of a transversal.
Identify the angles.
___________ ___________
http://www.mathwarehouse.com/geometry/angle/interactive-transveral-angles.php
(III) Transitive Property
If A = B and B = C then ________________
Transversal Line
Unit 6 Properties of Angles and Triangles 7
(IV) Proving Conjectures Involving Interior Angles Formed
by Parallel Lines
Alternate Interior Angles
In the diagram ∠1 = 120⁰ and line m and n are parallel
determine the value of:
∠2 = ________
∠3 = ________
What do you notice about
alternate interior angles?
____________________________
Make a conjecture about all alternate interior angles formed by a transversal that
intersect parallel lines.
_________________________________________________________________
_________________________________________________________________
Prove the conjecture based on the diagram given below.
Line m∥ n
m
n
∠1 = 120°
∠2
∠3
m
n
∠1
∠2
∠3
Statement Justification
Unit 6 Properties of Angles and Triangles 8
Alternate Exterior Angles
In the diagram ∠1 = 120⁰ and line m and n are parallel
determine the value of:
∠3 = ________
∠2 = ________
What do you notice about
alternate exterior angles?
____________________________
Make a conjecture about all alternate exterior angles formed by a transversal that
intersect parallel lines.
_________________________________________________________________
_________________________________________________________________
Prove the conjecture based on the diagram given below.
Line m∥ n
Statement Justification
m
n
∠1 = 120°
∠2
∠3
m
n
∠1
∠2
∠3
Unit 6 Properties of Angles and Triangles 9
Same Side Interior Angles
In the diagram ∠1 = 120⁰ and line m and n are parallel
determine the value of:
∠2 = ________
∠3 = ________
What do you notice about
sum of same side interior angles?
____________________________
Make a conjecture about all same side interior angles formed by a transversal that
intersect parallel lines.
_________________________________________________________________
_________________________________________________________________
Prove the conjecture based on the diagram given below.
Line m∥ n
Statement Justification
m
n
∠1 = 120°
∠2
∠3
m
n
∠1
∠2
∠3
Unit 6 Properties of Angles and Triangles 10
(V) Summary
When a Transversal intersects parallel lines,
●corresponding angles are equal
∠___ ≅ ∠___
∠___ ≅ ∠___
∠___ ≅ ∠___
∠___ ≅ ∠___
●alternate interior angles are equal
∠___ ≅ ∠___ and ∠___ ≅ ∠___
●alternate exterior angles are equal
∠___ ≅ ∠___ and ∠___ ≅ ∠___
●same side interior angles are supplementary
∠___ + ∠___ = 180⁰ and ∠___ + ∠___ = 180⁰
(VI) Reasoning to Determine Unknown Angles
Example: Determine the measures of:
∠a = _______
∠b = _______
∠c = _______
∠d = _______
110⁰
a b
c
d
Unit 6 Properties of Angles and Triangles 11
(VII) Using Angle Properties to Prove that Lines are Parallel
Example: Use the angle measures to
prove that braces CG, BF
and AE are parallel.
A
B
C
D
E
F
G
H
78⁰
78⁰
78⁰
78⁰
35⁰
35⁰
22⁰
22⁰
Questions: P.78 – 79 #1, #2, #3, #4, #8
Unit 6 Properties of Angles and Triangles 12
2.3 Angle Properties in Triangles
(I) Prove that the sum of all interior angles in a triangle is 180°.
Given: 𝐴𝐶 ⃡ || 𝐷𝐸 ⃡
Prove: 2 + 4 + 5 = 180°
What do I know? STATEMENT How do I know it? REASON
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
Applying the sum of interior angles in a triangle to determine an unknown angle.
Example: Determine the measure of A.
1 2 3
4 5
A B C
D E
6x - 30
-x + 50
3x A
B
C
Goals:
Prove properties of angles in triangles, and use these properties to solve problems.
Unit 6 Properties of Angles and Triangles 13
(II) Using Angle Sums to Determine Angle Measures
Example: Determine the measures of the unknown angles in ∆ABC.
(III) Non-Adjacent Interior Angles
●The TWO ANGLES of a triangle that do not have the same vertex as an interior angle.
(IV) Determining the Relationship Between the Exterior Angle and the Interior Angles
Example: Determine the measure
of 3.
What is the relationship between the exterior angle measure and the non-adjacent
interior angles?
_________________________________________________________________
In general,
_______________________________
40⁰
140⁰ A
B
C D
1
3 2
Exterior Angle
Non–Adjacent Interior Angles
44°
3 100°
1
3 2
Exterior Angle
Non–Adjacent Interior Angles
Unit 6 Properties of Angles and Triangles 14
Using Reasoning to Solve Problems
Example: Determine the unknown angles
1. 2.
3. 4.
5. 6.
x
Questions: P.90 – 93 #3, #5, #7, #10 - #16
Unit 6 Properties of Angles and Triangles 15
2.4 Angle Properties in Polygons
REMEMBER:
• The measure of an exterior angle of a triangle is equal to the sum of the
measures of the non-adjacent interior angles.
3 = 1 + 2
• The sum of the measure of the interior angles of a triangle is 180°.
(I) Determining the Relationship between the sum of interior angles and the number
of sides (n) in a polygon.
EXPLORE Part I Interior Angles
• A pentagon has three right angles and four sides of equal length, as shown.
What is the sum of the measures of the angles in the polygon?
1
3 2
Exterior Angle
Non–Adjacent Interior Angles
1
2
3
1 + 2 + 3 = 180°
Goals:
Determine properties of angles in polygons, and use these properties to solve problems.
Unit 6 Properties of Angles and Triangles 16
Create triangles within each polygon to determine the sum of measures of the
interior angles.
Polygon
Number of
Sides
Number of
Triangles
Sum of Angle
Measures
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Make a conjecture about the relationship between the sum of the measures of the
of the interior angles of a polygon.
________________________________________________________________
________________________________________________________________
EXPLORE Part II Exterior Angles
When the side of a polygon is extended, two
angles are created. The exterior angle is adjacent
to the interior angle.
Unit 6 Properties of Angles and Triangles 17
(II) Determining the Relationship between the number of sides in a polygon and the
sum of exterior angles.
NOTE: Convex Polygons
•A polygon in which each interior angle measures less than 180°.
For each polygon given below:
(a) determine all interior and exterior angles.
(b) determine the sum of exterior angles.
(i) (ii)
What do you notice about the sum of exterior angles around a convex polygon?
___________________________________________________________________
x°
30° 85°
y° 70°
z°
w°
x°
y°
75° 67°
116° z° 64°
w° v°
Unit 6 Properties of Angles and Triangles 18
A Regular Polygon
•is both equiangular and equilateral.
Example: The ends of the garbage box below forms a regular octagon.
Determine the measure of each interior angle.
The measure of each interior angle of a regular polygon is:
Problems:
1. Each interior angle of a convex polygon is 120°. Determine the number of sides
of the polygon.
2. The sum of the measures of interior angles of a polygon is 900°. Determine the
number of sides.
Questions: P.99-102 #1, #6, #7, #10, #13, #16
Unit 6 Properties of Angles and Triangles 19
PROPERTIES OF ANGLES AND TRIANGLES
INCLASS ASSIGNMENT REVIEW SHEET
Fill – in – the – Blank
Refer to the diagram below and answer the following questions.
1. True or False: ∠g is equal to ∠a. 1.______________________
2. Identify a pair of alternate interior angles. 2.______________________
3. Identify a pair of corresponding angles. 3.______________________
4. Identify a pair of same side interior angles. 4.
5. Identify a pair of vertically opposite angles. 5.
6. If ∠b is 700, what is the measure of ∠c? 6.______________________
7. If ∠a is 1200, what is the measure of ∠h? 7.______________________
8. State whether or not the following lines are parallel.
(a) (b) (c) (d)
(e) (f)
46 o
46 o 46 o
46 o 46 o
46 o
46 o
46 o
a b d c
e f g h
880 1020
320 320
Unit 6 Properties of Angles and Triangles 20
(a) (b) (c)
(d) (e) (f)
Multiple Choice
Choose the letter of the correct response.
9. Which pairs of angles are equal in this diagram?
10. Which statement about the angles in this diagram is false?
A) e = f B) f = a C) a = b
D) d = c
11. In which diagrams are two lines parallel?
1.
2.
3.
A) Choice 1 only C) Choices 1, 2, and 3
B) Choice 1 and Choice 3 D) Choice 2 and Choice 3
A) b = c, e = g, and f = h C) b = e, c = h, and d = g
B) b = f, c = g, and d = h D) b = a, c = e, and d = f
Unit 6 Properties of Angles and Triangles 21
12. Which angle property proves EFS = 28°?
A) alternate exterior angles C) alternate interior angles
B) supplementary angles D) corresponding angles
13. Which is the value of x in order for the two lines to be parallel?
(A) 45o (B) 55o (C) 125o (D) 305o
14. Which represents the value of x?
(A) 82o (B) 127o (C) 45o (D) 90o
x
45o 127o
x
55 o
Unit 6 Properties of Angles and Triangles 22
15. Which are the correct measures of the indicated angles?
A) w = 116°, x =64°, y = 64° C) w = 134°, x = 46°, y = 46°
B) w = 136°, x = 44°, y = 136° D) w = 146°, x = 44°, y = 146°
16. Which are the correct measures of the interior angles of CDE?
A) DCE = 56°, CDE = 101°, and CED = 23°
B) DCE = 46°, CDE = 101°, and CED = 33°
C) DCE = 32°, CDE = 83°, and CED = 65°
D) DCE = 76°, CDE = 91°, and CED = 13°
17. Determine the sum of the measures of the angles in a 16-sided convex polygon.
A) 2340° B) 2880° C) 2520° D) 2700°
18. Determine the measure of one interior angle in a 12-sided convex polygon.
A) 1800° B) 150° C) 2160° D) 180°
Unit 6 Properties of Angles and Triangles 23
CONSTRUCTED RESPONSE QUESTIONS
19. Complete the following proof.
Given: WV || YX
Prove: UTY = VST
Z
Statement Reason
WV || YX
USV = WST
USV = STX
WST = STX
20. Determine the value of x for each of the following diagrams.
(a) (b)
(c) (d)
Y X
W V
U
S
T
Unit 6 Properties of Angles and Triangles 24
(e) (f)
21. Determine the measures of the unknown angles for each of the following diagrams.
(a) (b)
(c) (d)
x°
y°
x 100° z
40°
y
Unit 6 Properties of Angles and Triangles 25
22. Determine the value of x and the measures of both DOG and DGM.
23. Determine the sum of all the interior angles and determine the measure of one interior angle.
24. The sum of the measures of the interior angles of an unknown polygon is 1980o.
Determine the number of sides of this polygon.
D
O G
55°
4x 6x + 15
M
Unit 6 Properties of Angles and Triangles 26
SOLUTIONS
1. False 2. c = f, b = g 3. a = c, b = d, e = g, f = h 4. b and c, f and g 5. a = f, b = e, c = h, d = g
6. c = 110° 7. h = 120°
8(a) parallel (b) not parallel (c) not parallel (d) parallel (e) not parallel (f) parallel
9. B 10. B 11. D 12. A 13. C 14. A 15. C 16. A 17. C 18. B
19.
Statement Reason
WV || YX Given
USV = WST Vertically Opposite Angles (X)
USV = STX Corresponding Angles (F)
WST = STX Transitive Property
20(a) x = 35 (b) x = – 7 (c) x = 7 (d) x = 8 (e) x = 4 (f) x = 12
21(a) p = 80°, q = 130°, r = 50° (b) x = 120°, y = 60°
(c) w = 65°, x = 85°, y = 95°, z = 115° (d) x = 80°, y = 40°, z = 60°
22. x = 11, DOG = 81°, DGM = 136°
23. sum of all interior angles = 720° , sum of one interior angle = 120°
24. n = 13 sides
Unit 6 Properties of Angles and Triangles 27
2.5 Exploring Congruent Triangles
What pieces of information do we need to prove that all of the
triangular roof trusses are congruent?
Whenever two figures have the same _________ and _________
then the figures are congruent.
(I) Congruent Triangles
Are the triangles below congruent?
State the triangles that are congruent. ∆____ ≅ ∆____
State the corresponding sides and corresponding angles that are congruent.
Corresponding Sides Corresponding Angles
___ ___ __ __
___ ___ __ __
___ ___ __ __
A B
C
3
4
5
F
E D
3
4
5
Goals:
Determine the minimum amount of information needed to prove two triangles congruent.
Unit 6 Properties of Angles and Triangles 28
(II) Ways to Prove Triangles Congruent
Side – Side – Side (SSS Postulate)
•If three sides of one triangle are congruent to three sides of another triangle
then the triangles are congruent.
Example: State the congruent triangles.
Side – Angle – Side (SAS Postulate)
•If two sides and the included angle of one triangle are congruent to two sides
and the included angle of another triangle then the triangles are congruent.
Example: State the congruent triangles.
Angle – Side – Angle (ASA Postulate)
•If two angles and the included side of one triangle are congruent to two angles
and the included side of another triangle then the triangles are congruent.
Example: State the congruent triangles.
P
Q
R S
13 13
5 5
W
V
X
Y
4
4
2
2
Z
G
F J
H
Unit 6 Properties of Angles and Triangles 29
Angle – Angle – Side (AAS Postulate)
•If two angles and a non-included side of one triangle are congruent to two angles
and a non-included side of another triangle then the triangles are congruent.
Example: State the congruent triangles.
Example: State the appropriate congruence postulate SSS, SAS, ASA or AAS
if the triangles are congruent.
5)
P
R S Q
T
Questions: P.106 #1, #2, #3
Unit 6 Properties of Angles and Triangles 30
2.6 Proving Congruent Triangles
REMEMBER: Ways to prove triangles congruent
NOTE: When two triangles are congruent, all 6 corresponding parts are congruent.
Side – Side – Side (SSS Postulate)
•If three sides of one triangle are
congruent to three sides of another
triangle then the triangles are
congruent.
Angle–Side–Angle (ASA Postulate)
•If two angles and the included side of
one triangle are congruent to two
angles and the included side of another
triangle, then the triangles are
congruent.
Side–Angle–Side (SAS Postulate)
•If two sides and the included angle of
one triangle are congruent to two sides
and the included angle of another
triangle, then the triangles are
congruent.
Angle–Angle–Side (AAS Postulate)
•If two angles and a non-included side
of one triangle are congruent to two
angles and a non-included side of
another triangle, then the triangles are
congruent.
Goals:
Using deductive reasoning to prove that triangles are congruent.
Unit 6 Properties of Angles and Triangles 31
(I) Using reasoning in a two column proof to prove triangles congruent
Example:
Given: RS || TU
𝑆𝑉̅̅̅̅ ≅ 𝑇𝑉̅̅ ̅̅
Prove: ∆𝑅𝑆𝑉 ≅ ∆𝑈𝑇𝑉
What do I know? STATEMENT How do I know it? REASON
1.
1.
2.
2.
3.
3.
4.
4.
(II) Using congruent triangles to deduce two segments or two angles are congruent by
corresponding parts.
Example:
Given: 𝑅𝑃̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑄𝑃𝑆
𝑃𝑄̅̅ ̅̅ ≅ 𝑃𝑆̅̅̅̅
Prove: 𝑅𝑄̅̅ ̅̅ ≅ 𝑅𝑆̅̅̅̅
What do I know? STATEMENT How do I know it? REASON
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
S
R
V
T
U
S
P
Q R
Unit 6 Properties of Angles and Triangles 32
Example:
Given: 𝑇𝑃̅̅̅̅ AC̅̅̅̅
𝐴𝑃̅̅ ̅̅ ≅ 𝐶𝑃̅̅ ̅̅
Prove: ∆𝑇𝐴𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠
What do I know? STATEMENT How do I know it? REASON
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6. 6.
7.
7.
A
T
C P
Questions: P.112-115 #1, #2, #4 – #7, #12 – #15
Unit 6 Properties of Angles and Triangles 33
TEST REVIEW SHEET
1. What is the relationship between ∠𝑤 and ∠𝑦? 1.____
(A) Alternate Interior Angles
(B) Corresponding Angles
(C) Same Side Interior Angles
(D) Vertically Opposite Angles
2. Given two parallel lines and a transversal, which pair of angles are equal? 2.____
(A) A = C , B = D
(B) A = E , D = H
(C) C = E , D = F
(D) C = D , G = H
3. Which figure illustrates that the two lines are NOT parallel given the two angle measures? 3.
(A) Figure 1 (B) Figure 2 (C) Figure 3 (D) Figure 4
138o 35 o
o 35 o
32o
o
32o
148 o
148 o
42 o
o
A B
C D
E F G H
Unit 6 Properties of Angles and Triangles 34
4. Given the two parallel lines, determine the measure of x. 4.
(A) x = 125º
(B) x = 135º
(C) x = 45º
(D) x = 55º
5. Given the two parallel lines, determine the value of x. 5.
(A) 30o (B) 50o (C) 130o (D) 150o
6. Determine the value of x. 6.
(A) 34° (B) 146°
(C) 35° (D) 145°
7. What are the correct measures of the indicated measures? 7.
(A) x = 60° , y = 60° , z = 120°
(B) x = 60° , y = 120° , z = 60°
(C) x = 120° , y = 120° , z = 60°
(D) x = 120° , y = 60° , z = 120°
x
150°
125º
x
x
y
z
120°
x
34° 35°
Unit 6 Properties of Angles and Triangles 35
8. Determine the measure of x. 8.____
(A) x = 40º
(B) x = 140º
(C) x = 105º
(D) x = 75º
9. Determine the value of x. 9.
(A) x = 10° (B) x = 20° (C) x = 30° (D) x = 40°
10. Determine the value of x. 10.
(A) x = 5° (B) x = 15° (C) x = 10° (D) x = 30°
11. Determine the measure of A. 11.
(A) 80° (B) 60°
(C) 40° (D) 20°
4x + 15o
x + 45o
4x + 20o
2x + 60o
x
75°
65°
C
A
B
2x
3x
4x
Unit 6 Properties of Angles and Triangles 36
12. Determine the value of x. 12.
(A) x = 10°
(B) x = 20°
(C) x = 40°
(D) x = 60°
13. Which represents the value of x? 13.
(A) 74o (B) 64o (C) 121o (D) 59o
14. What is the sum of the measures of all the angles in a regular decagon (ten sided figure)? 14.
(A) 1800° (B) 144° (C) 180° (D) 1440°
15. What is the measure of one interior angle in a regular hexagon (six sided figure)? 15.
(A) 1080° (B) 720° (C) 180° (D) 120°
16. How many sides are there in a convex polygon that has the sum of all its interior angles 16.
equal to 1260° ?
(A) 10 sides (B) 9 sides (C) 8 sides (D) 7 sides
x
47o
121o
120º 2x
x
Unit 6 Properties of Angles and Triangles 37
17. Which additional piece of information would allow you to conclude that these triangles 17.
are congruent?
(A) AC = DF (B) C = F (C) AB = EF (D) BC = EF
18. What can you deduce from the congruence statement ABC DEF ? 18.
(A) AB = EF (B) AC = EF (C) BC = DE (D) AC = DF
19. What can you deduce from the congruence statement ABC PQR ? 19.
(A) A = R (B) B = P (C) C = R (D) C = Q
20. Which congruence postulate shows that ABC XYZ? 20.
(A) Side – Side – Side Postulate (B) Angle – Side – Angle Postulate
(C) Angle – Angle – Side Postulate (D) Side – Angle – Side Postulate
21. Which piece of information is required to prove that ABC DCB using the 21.
SAS postulate ?
(A) 𝐴𝐵̅̅ ̅̅ = 𝐷𝐶̅̅ ̅̅
(B) 𝐵𝐶̅̅ ̅̅ = 𝐶𝐵̅̅ ̅̅
(C) 𝐴𝐶̅̅ ̅̅ = 𝐷𝐵̅̅ ̅̅
(D) 𝐴𝐵̅̅ ̅̅ = 𝐷𝐵̅̅ ̅̅ C D
A B
Unit 6 Properties of Angles and Triangles 38
CONSTRUCTED RESPONSE QUESTIONS
22. Determine the value of x.
23. Determine the value of x AND then determine the measures of both DOG and DGM.
24. Determine the value of x and the measures of BCD and CDB.
25. Determine the value of x for each of the following diagrams.
(b) (b)
O G
2x – 5°
D
3x + 45°
80°
M
4x + 28o
2x + 32o
110º x + 20°
3x + 30°
B D
C
x + 16°
78° x + 24°
2x + 50º
5x + 14°
Unit 6 Properties of Angles and Triangles 39
(c)
26. Determine the measure of the missing variables for the following diagram.
(a)
27. Determine the measures of the missing variables for the following diagrams.
(a) (b)
(c) (d)
w p 115° q
20º
x + 55°
3x + 25°
x
84°
62°
y z
80º
45° a b
c
d
e
60°
f
110⁰
a b
c
d x 100°
z 40°
y
Unit 6 Properties of Angles and Triangles 40
(e)
28(a) Determine the measure of one interior angle in the regular octagon below.
(b) The sum of the measures of the interior angles of an unknown polygon is 1980o.
Determine the number of sides of this polygon.
(c) The sum of the measures of all the interior angles of an unknown polygon is 1620o.
Determine the number of sides in the unknown polygon.
Unit 6 Properties of Angles and Triangles 41
29. Complete the following proof.
Given: WV || YX
Prove: USV = STX
Z
Statement Reason
WV || YX
WST = USV
WST = STX
USV = STX
30. Name the congruence postulate ( SSS, SAS, ASA, or AAS ) and give the congruence statement
for the triangles.
(a) (b)
Y X
W V
U
S
T
F
C
D
B
E M K L
N
Unit 6 Properties of Angles and Triangles 42
31. Complete the following proof.
Given: PQ || RS
R = Q = 90º
Prove: SQ = PR
STATEMENT
REASON
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
Q P
R S
Unit 6 Properties of Angles and Triangles 43
32. Given: PR SQ
RS = RQ
Prove: S = Q
STATEMENT REASON
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
S
P
Q R
Unit 6 Properties of Angles and Triangles 44
SOLUTIONS
1. B 2. B 3. C 4. D 5. D 6. D 7. A 8. B 9. B 10. C 11. B 12. C 13. A 14. D
15. D 16. B 17. D 18. D 19. C 20. A 21. B
22. x = 20 23. x = 30, DOG = 55°, DGM = 135° 24. x = 15, BCD = 75°, CDB = 35°
25(a) x = 31 (b) x = 12 (c) x = 25 26. a = 45°, b = 55°, c = 55°, d = 135°, e = 45°, f = 65°
27(a) x = 34°, y = 62°, z = 34° (b) p = 65°, q = 50°, w = 65° (c) x = 80°, y = 40°, z = 60°
27(d) a = 110°, b = 110°, c = 70°, d = 70° (e) p = 80°, q = 130°, r = 50°
28(a) sum = 135° (b) n = 13 sides (c) n = 11 sides
29.
Statement Reason
WV || YX Given
WST = USV Vertically Opposite Angles (X)
WST = STX Alternate Interior Angles (Z)
USV = STX Transitive Property
30(a) ASA postulate, BCD FED (b) SAS postulate, NLK NLM
31.
STATEMENT REASON
1. PQ || RS 1. Given
2. R = Q = 90º
2. Given
3. QPS = RSP 3. Alternate Interior Angles (Z)
4. PS = PS 4. Common Side
5. SQP PRS 5. AAS
6. SQ = PR 6. Definition of Congruent Triangles
Unit 6 Properties of Angles and Triangles 45
32.
STATEMENT REASON
1. PR SQ 1. Given
2. SRP = QRP 2. Both angles equal 90°
3. RS = RQ 3. Given
4. PR = PR 4. Common Side
5.SRP QRP 5. SAS
6. S = Q 6. Definition of Congruent Triangles
