GEOMETRY - whiteplainspublicschools.org · DAY 6: (Ch. 4-5) SWBAT: Prove ... SAS, ASA, AAS, and HL....
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Transcript of GEOMETRY - whiteplainspublicschools.org · DAY 6: (Ch. 4-5) SWBAT: Prove ... SAS, ASA, AAS, and HL....
GEOMETRY
Chapter 4: Triangles
Name:_____________________________
Teacher:____________________________
Pd: _______
Table of Contents
DAY 1: (Ch. 4-1 & 4-2) SWBAT: Classify triangles by their angle measures and side lengths. Pgs: 1-5 Use triangle classification to find angle measures and side lengths. Pgs: 6-7
DAY 2: (Ch. 4-2) SWBAT: Apply theorems about the interior and exterior angles of triangles. Pgs: 8-12 HW: Pgs: 13-15
DAY 3: (Review) SWBAT: Use triangle classification to find angle measures and side lengths. Pgs: 16-18 Apply theorems about the interior and exterior angles of triangles.
DAY 3: Take Home Quiz: Day 1 to DAY 3
DAY 4: (Ch. 4-3) SWBAT: Use properties of congruent triangles. Pgs: 19-24 Prove triangles congruent by using the definition of congruence. HW: Pgs: 25-27
DAY 5: (Ch. 4-4) SWBAT: Prove triangles congruent by using SSS and SAS. Pgs: 28-33 HW: Pgs: 34-35
DAY 6: (Ch. 4-5) SWBAT: Prove triangles congruent by using ASA and AAS. Pgs: 36-40 HW: Pgs: 41-42
DAY 7: (Ch. 4-5) SWBAT: Prove triangles congruent by using HL. Pgs: 43-46 HW: Page 47
DAY 8: (Review) SWBAT: Prove triangles congruent by using SSS, SAS, ASA, AAS, and HL. Pgs: 48-50
1
Day 1: Triangle Vocabulary and Theorems
Warm – Up
Part I: Vocabulary. Fill out the following chart below.
Example 1:
7
Find the measure of each angle of the triangle.
11. 12.
13. The angle measures of a triangle are in the ratio of 5:6:7. Find the angle measures of the triangle.
14. The angle measures of a triangle are in the ratio of 10:12:14. Find the angle measures of the triangle.
15. If the measures, in degrees, of the three angles of a triangle are x, x + 10, and 2x − 6, the
triangle must be:
1) Isosceles
2) Equilateral
3) Right
4) Scalene
9
Part I: Angle relationships in triangles. Find the measure of all angles in the triangles below.
Then answer the following questions and try to develop the theorems that represent these relationships.
After checking the theorems with your teacher, then complete the remaining examples.
a)
b) c)
10
Part II: Conclusions
1. Investigate the Triangle Sum Theorem and its corollaries
a) 62 + 71 + _____ = _____ (m
b) 23 + 27 + _____ = _____ (m
c) 90 + 37 + _____ = _____ (m
2. Investigate the Exterior Angles Theorem
a) 62 + 71 = _____ m b) 23 + 27 = _____ m c) 37 + ______ = _____ m
(m
What relationship do you notice?
In any triangle, the sum of the interior angles is equal to ___________
In a right triangle, the two acute angles are _________________.
In an equiangular triangle, all angles measure ___________
The exterior angle of a triangle is always equal to
Formula: ____ + _____ = ______
11
Part III: Practice. Apply the new theorems to solve each problem
1. Solve for x.
2. Solve for m
3.
4. Ghfhfh
12
Challenge
Use the information given in the diagram to determine the m .
SUMMARY
2x2+3x-2
4x+3
x2+1
A D
B
C
18
13. If the measures of the angles of a triangle are in the ratio 1:3:5, the number of degrees in the measure of the smallest angle is….
14.
15. ACD is an exterior angle of ABC, m A = 3x, m ACD = 5x, and m B = 50. What is the value of x?
20
Geometric figures are congruent if they are the same size and shape. Corresponding angles and
corresponding sides are in the same _______________ in polygons with an equal number of _______.
Two polygons are _________ polygons if and only if their _________________ sides are _____________. Thus
triangles that are the same size and shape are congruent.
Ex 1: Name all the corresponding sides and angles below if
Corresponding Sides Corresponding Angles
21
Ex 2:
In a congruence statement, the order of the vertices indicates the corresponding parts.
Ex 3: If ∆PQR ∆STW, identify all pairs of corresponding congruent parts.
Corresponding Sides Corresponding Angles
28
Day 4 – SSS AND SAS Methods of Proving Triangles Congruent
Warm-Up
Five Ways to Prove Triangles Congruent
In the previous lesson, you learned that congruent triangles have all corresponding sides and all
corresponding angles congruent. Do we need to show all six parts congruent to conclude that two
triangles are congruent?
The answer is no. We can show triangles are congruent by showing few than all three sides and angles
congruent, so long as these congruent sides and angles are in the correct order. The arrangements
that prove triangles congruent are as follows:
Side-Side-Side (SSS)
Side-Angle-Side (SAS)
Angle-Side-Angle (ASA)
Angle-Angle-Side (AAS)
Hypotenuse-Leg (HL) – for right triangles only
We will take a look at each of these in turn. Today we are going to focus on (SSS) and (SAS).
29
Mark the triangles below to prove the triangles are congruent by the SSS
Theorem.
Example 1:
You Try!
Example 2:
You Try!
G
X
Z
Y
G
X
Z
Y
31
Mark the triangles below to prove the triangles are congruent by the SAS
Theorem.
Example 4: Given:
Prove: ∆ABC ∆ZXY
Example 5: Given:
Prove:
You Try! Given:
Prove:
36
Day 5 – ASA AND AAS Methods of Proving Triangles Congruent
Warm-Up
For # 1 - 3, state if there is enough information to prove the triangles are congruent. If there is,
state the theorem used and write the congruency statement.
1.
2.
3. f
37
Mark the triangles below to prove the triangles are congruent by the ASA Theorem.
Example 1:
Try It!
Try It!:
39
Mark the triangles below to prove the triangles are congruent by the AAS Theorem.
Example 2:
You Try It!
You Try it!
44
Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell
what other information is needed.
Examples:
_____________________ _____________________
Practice
Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell
what other information is needed.
45
Mark the triangles below to determine whether the HL theorem can be used to prove
triangles congruent.
4.
You Try!
Given: , , and
P Q
R S