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Transcript of Chapter4001 and 4002 traingles
Holt McDougal Geometry
4-1 Classifying Triangles
Hon Geom Drill 12/11/14
• Take out any hw you have and then
complete the drill on your own.
Use your homework to help you
answer the questions. There is a
front and back
Holt McDougal Geometry
4-1 Classifying Triangles
Warm Up
Classify each angle as acute, obtuse, or right.
3. 4.
5.
6. If the perimeter is 47, find x and the lengths
of the three sides.
rightacute
x = 5; 8; 16; 23
obtuse
1. Take a look at the three triangles I have provided
and answer the following questions:
What do you notice?
What do you wonder?
2. List any information you already know about triangles, their angles, and
classify triangles
Holt McDougal Geometry
4-1 Classifying Triangles
1. Classify triangles by their angle measures and side lengths.
2. Use triangle classification to find angle
measures and side lengths.
3. Find the measures of interior and exterior angles of triangles.
4. Apply theorems about the interior
and exterior angles of triangles.
Objectives
Holt McDougal Geometry
4-1 Classifying Triangles
acute triangle
equiangular triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Vocabulary
Holt McDougal Geometry
4-1 Classifying Triangles
auxiliary line
corollary
interior
exterior
interior angle
exterior angle
remote interior angle
Vocabulary
Holt McDougal Geometry
4-1 Classifying Triangles
Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths.
Holt McDougal Geometry
4-1 Classifying Triangles
B
A
C
AB, BC, and AC are the sides of ABC.
A, B, C are the triangle's vertices.
Holt McDougal Geometry
4-1 Classifying Triangles
Acute Triangle
Three acute angles
Triangle Classification By Angle Measures
Holt McDougal Geometry
4-1 Classifying Triangles
Equiangular Triangle
Three congruent acute angles
Triangle Classification By Angle Measures
Holt McDougal Geometry
4-1 Classifying Triangles
Right Triangle
One right angle
Triangle Classification By Angle Measures
Holt McDougal Geometry
4-1 Classifying Triangles
Obtuse Triangle
One obtuse angle
Triangle Classification By Angle Measures
Holt McDougal Geometry
4-1 Classifying Triangles
Classify BDC by its angle measures.
Example 1A: Classifying Triangles by Angle Measures
DBC is an obtuse angle.
DBC is an obtuse angle. So BDC is an obtuse triangle.
Holt McDougal Geometry
4-1 Classifying Triangles
Classify ABD by its angle measures.
Example 1B: Classifying Triangles by Angle Measures
ABD and CBD form a linear pair, so they are supplementary.
Therefore mABD + mCBD = 180°. By substitution, mABD + 100° = 180°. So mABD = 80°. ABD is an acute triangle by definition.
Holt McDougal Geometry
4-1 Classifying Triangles
Classify FHG by its angle measures.
Check It Out! Example 1
EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°.
FHG is an equiangular triangle by definition.
Holt McDougal Geometry
4-1 Classifying Triangles
Equilateral Triangle
Three congruent sides
Triangle Classification By Side Lengths
Holt McDougal Geometry
4-1 Classifying Triangles
Isosceles Triangle
At least two congruent sides
Triangle Classification By Side Lengths
Holt McDougal Geometry
4-1 Classifying Triangles
Scalene Triangle
No congruent sides
Triangle Classification By Side Lengths
Holt McDougal Geometry
4-1 Classifying Triangles
Remember!
When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.
Holt McDougal Geometry
4-1 Classifying Triangles
Classify EHF by its side lengths.
Example 2A: Classifying Triangles by Side Lengths
From the figure, . So HF = 10, and EHF is isosceles.
Holt McDougal Geometry
4-1 Classifying Triangles
Classify EHG by its side lengths.
Example 2B: Classifying Triangles by Side Lengths
By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHGis scalene.
Holt McDougal Geometry
4-1 Classifying Triangles
Classify ACD by its side lengths.
Check It Out! Example 2
From the figure, . So AC = 15, and ACD is isosceles.
Holt McDougal Geometry
4-1 Classifying Triangles
Find the side lengths of JKL.
Example 3: Using Triangle Classification
Step 1 Find the value of x.
Given.
JK = KL Def. of segs.
4x – 10.7 = 2x + 6.3Substitute (4x – 10.7) for JK and (2x + 6.3) for KL.
2x = 17.0
x = 8.5
Add 10.7 and subtract 2x from both sides.
Divide both sides by 2.
Holt McDougal Geometry
4-1 Classifying Triangles
Find the side lengths of JKL.
Example 3 Continued
Step 2 Substitute 8.5 into the expressions to find the side lengths.
JK = 4x – 10.7
= 4(8.5) – 10.7 = 23.3
KL = 2x + 6.3
= 2(8.5) + 6.3 = 23.3
JL = 5x + 2
= 5(8.5) + 2 = 44.5
Holt McDougal Geometry
4-1 Classifying Triangles
Find the side lengths of equilateral FGH.
Check It Out! Example 3
Step 1 Find the value of y.
Given.
FG = GH = FH Def. of segs.
3y – 4 = 2y + 3
Substitute (3y – 4) for FG and (2y + 3) for GH.
y = 7Add 4 and subtract 2y from both sides.
Holt McDougal Geometry
4-1 Classifying Triangles
Find the side lengths of equilateral FGH.
Check It Out! Example 3 Continued
Step 2 Substitute 7 into the expressions to find the side lengths.
FG = 3y – 4
= 3(7) – 4 = 17
GH = 2y + 3
= 2(7) + 3 = 17
FH = 5y – 18
= 5(7) – 18 = 17
Holt McDougal Geometry
4-1 Classifying Triangles
The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.
P = 3(18)
P = 54 ft
A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?
Example 4: Application
Holt McDougal Geometry
4-1 Classifying Triangles
A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam?
Example 4: Application Continued
To find the number of triangles that can be made from 420 feet of steel beam, divide 420 by the amount of steel needed for one triangle.
420 54 = 7 triangles 7 9
There is not enough steel to complete an eighth triangle. So the steel mill can make 7 triangles from a 420 ft. piece of steel beam.
Holt McDougal Geometry
4-1 Classifying Triangles
Holt McDougal Geometry
4-1 Classifying Triangles
An auxiliary line is a line that is added to a figure to aid in a proof.
An auxiliary line used in the Triangle Sum
Theorem
Holt McDougal Geometry
4-1 Classifying Triangles
After an accident, the positions of cars are measured by law
enforcement to investigate the collision. Use the diagram
drawn from the information collected to find mXYZ.
Example 1A: Application
mXYZ + mYZX + mZXY = 180° Sum. Thm
mXYZ + 40 + 62 = 180Substitute 40 for mYZX and
62 for mZXY.
mXYZ + 102 = 180 Simplify.
mXYZ = 78° Subtract 102 from both sides.
Holt McDougal Geometry
4-1 Classifying Triangles
After an accident, the positions of cars are measured by law
enforcement to investigate the collision. Use the diagram
drawn from the information collected to find mYWZ.
Example 1B: Application
mYXZ + mWXY = 180° Lin. Pair Thm. and Add. Post.
62 + mWXY = 180 Substitute 62 for mYXZ.
mWXY = 118° Subtract 62 from both sides.
Step 1 Find mWXY.
118°
Holt McDougal Geometry
4-1 Classifying Triangles
After an accident, the positions of cars are measured by law
enforcement to investigate the collision. Use the diagram drawn from the information collected
to find mYWZ.
Example 1B: Application Continued
Step 2 Find mYWZ.
118°
mYWX + mWXY + mXYW = 180° Sum. Thm
mYWX + 118 + 12 = 180 Substitute 118 for mWXY and 12 for mXYW.
mYWX + 130 = 180 Simplify.
mYWX = 50° Subtract 130 from both sides.
Holt McDougal Geometry
4-1 Classifying Triangles
A corollary is a theorem whose proof follows directly from another theorem. Here are two
corollaries to the Triangle Sum Theorem.
Holt McDougal Geometry
4-1 Classifying Triangles
One of the acute angles in a right triangle measures 2x°. What is the measure of the other
acute angle?
Example 2: Finding Angle Measures in Right Triangles
mA + mB = 90°
2x + mB = 90 Substitute 2x for mA.
mB = (90 – 2x)° Subtract 2x from both sides.
Let the acute angles be A and B, with mA = 2x°.
Acute s of rt. are comp.
Holt McDougal Geometry
4-1 Classifying Triangles
The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of
the other acute angle?
Check It Out! Example 2a
mA + mB = 90°
63.7 + mB = 90 Substitute 63.7 for mA.
mB = 26.3° Subtract 63.7 from both sides.
Let the acute angles be A and B, with mA = 63.7°.
Acute s of rt. are comp.
Holt McDougal Geometry
4-1 Classifying Triangles
The measure of one of the acute angles in a right triangle is x°. What is the measure of the
other acute angle?
Check It Out! Example 2b
mA + mB = 90°
x + mB = 90 Substitute x for mA.
mB = (90 – x)° Subtract x from both sides.
Let the acute angles be A and B, with mA = x°.
Acute s of rt. are comp.
Holt McDougal Geometry
4-1 Classifying Triangles
The measure of one of the acute angles in a right
triangle is 48 . What is the measure of the other
acute angle?
Check It Out! Example 2c
mA + mB = 90° Acute s of rt. are comp.
2°5
Let the acute angles be A and B, with mA = 48 . 2°5
Subtract 48 from both sides.2 5
Substitute 48 for mA.2 548 + mB = 90
2 5
mB = 41 3°5
Holt McDougal Geometry
4-1 Classifying Triangles
The interior is the set of all points inside the figure. The exterior is the set of all points
outside the figure.
Interior
Exterior
Holt McDougal Geometry
4-1 Classifying Triangles
An interior angle is formed by two sides of a triangle. An exterior angle is formed by one
side of the triangle and extension of an adjacent side.
Interior
Exterior
4 is an exterior angle.
3 is an interior angle.
Holt McDougal Geometry
4-1 Classifying Triangles
Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.
Interior
Exterior
3 is an interior angle.
4 is an exterior angle.
The remote interior angles of 4 are 1
and 2.
Find the missing angle measures.
Holt McDougal Geometry
4-1 Classifying Triangles
Holt McDougal Geometry
4-1 Classifying Triangles
Find mB.
Example 3: Applying the Exterior Angle Theorem
mA + mB = mBCD Ext. Thm.
15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.
2x + 18 = 5x – 60 Simplify.
78 = 3xSubtract 2x and add 60 to
both sides.
26 = x Divide by 3.
mB = 2x + 3 = 2(26) + 3 = 55°
Holt McDougal Geometry
4-1 Classifying Triangles
Find mACD.
Check It Out! Example 3
mACD = mA + mB Ext. Thm.
6z – 9 = 2z + 1 + 90 Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.
6z – 9 = 2z + 91 Simplify.
4z = 100Subtract 2z and add 9 to both
sides.
z = 25 Divide by 4.
mACD = 6z – 9 = 6(25) – 9 = 141°
Holt McDougal Geometry
4-1 Classifying Triangles
Holt McDougal Geometry
4-1 Classifying Triangles
Find mK and mJ.
Example 4: Applying the Third Angles Theorem
K J
mK = mJ
4y2 = 6y2 – 40
–2y2 = –40
y2 = 20
So mK = 4y2 = 4(20) = 80°.
Since mJ = mK, mJ = 80°.
Third s Thm.
Def. of s.
Substitute 4y2 for mK and 6y2 – 40 for mJ.
Subtract 6y2 from both sides.
Divide both sides by -2.
Holt McDougal Geometry
4-1 Classifying Triangles
Check It Out! Example 4
Find mP and mT.
P T
mP = mT
2x2 = 4x2 – 32
–2x2 = –32
x2 = 16
So mP = 2x2 = 2(16) = 32°.
Since mP = mT, mT = 32°.
Third s Thm.
Def. of s.
Substitute 2x2 for mP and 4x2 – 32 for mT.
Subtract 4x2 from both sides.
Divide both sides by -2.
Holt McDougal Geometry
4-1 Classifying Triangles
Lesson Quiz
Classify each triangle by its angles and sides.
1. MNQ
2. NQP
3. MNP
4. Find the side lengths of the triangle.
acute; equilateral
obtuse; scalene
acute; scalene
29; 29; 23