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4.1 Triangles and Angles

Warm Up

Classify each angle as acute, obtuse, or right.

1.2.

3.

4. If the perimeter is 47, find x and the lengths of the three sides.

right

acute

obtuse

x = 5; 8; 16; 23

4.1 Triangles and Angles

Objectives

Classify triangles by their angle measures and side lengths.

Use triangle classification to find angle measures and side lengths.Find the measures of interior and exterior angles of triangles.

Apply theorems about the interior and exterior angles of triangles.

4.1 Triangles and Angles

Vocabulary

acute triangle Corollary

equiangular triangle Legs

right triangle Adjacent

obtuse triangle Exterior

equilateral triangle Interior

isosceles triangle Hypotenuse

scalene triangle base

4.1 Triangles and Angles

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.

4.1 Triangles and Angles

C

A

AB, BC, and AC are the sides of ABC.

B

A, B, C are the triangle's vertices.

4.1 Triangles and Angles

In a right triangle, the two sides making the right angle are called Legs.

The side opposite of the right angle is called the Hypothenuse.

4.1 Triangles and Angles

In an isosceles triangle, the two sides that are congruent are called the Legs.

The third side is the called the base.

4.1 Triangles and Angles

B is an obtuse angle. So BDC is an obtuse triangle.

Example 1A: Classifying Triangles by Angle Measures

Classify BDC by its angle measures.

B is an obtuse angle.

4.1 Triangles and Angles

Therefore mABD + mCBD = 180°. By substitution, mABD + 100°= 180°. SomABD = 80°. ABD is an acute triangle by definition.

Example 1B: Classifying Triangles by Angle Measures

Classify ABD by its angle measures.

ABD andCBD form a linear pair, so they are supplementary.

4.1 Triangles and Angles

FHG is an equiangular triangle by definition.

Check It Out! Example 1

Classify FHG by its angle measures.

EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. SomFHG = 60°.

4.1 Triangles and Angles

Remember!

When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent.

4.1 Triangles and Angles

When the sides of the triangle are extended, the original angles are the interior angles.

The angles that are adjacent (next to) the interior angles are the exterior angles.

4.1 Triangles and Angles

4.1 Triangles and Angles

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

4.1 Triangles and Angles

4.1 Triangles and Angles

A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

4.1 Triangles and Angles

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

4.1 Triangles and Angles

Sum. Thm

Example 1A: Application

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.

mXYZ + mYZX + mZXY = 180°

Substitute 40 for mYZX and 62 for mZXY.

mXYZ + 40+ 62= 180

mXYZ + 102= 180

Simplify.

mXYZ = 78°

Subtract 102 from both sides.

4.1 Triangles and Angles

118°

Example 1B: Application

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.

Step 1 Find mWXY.

mYXZ + mWXY = 180°

Lin. Pair Thm. and Add. Post.

62 + mWXY = 180

Substitute 62 for mYXZ.

mWXY = 118°

Subtract 62 from both sides.

118°

Sum. Thm

Example 1B: Application Continued

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.

Step 2 Find mYWZ.

mYWX + mWXY + mXYW = 180°

Substitute 118 for mWXY and 12 for mXYW.

mYWX + 118+ 12= 180

mYWX + 130= 180

Simplify.

Subtract 130 from both sides.

mYWX = 50°

4.1 Triangles and Angles

Sum. Thm

Check It Out! Example 1

Use the diagram to find mMJK.

mMJK + mJKM + mKMJ = 180°

Substitute 104 for mJKM and 44 for mKMJ.

mMJK + 104+ 44= 180

mMJK + 148= 180

Simplify.

Subtract 148 from both sides.

mMJK = 32°

4.1 Triangles and Angles

A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

4.1 Triangles and Angles

Acute s of rt. are comp.

Example 2: Finding Angle Measures in Right Triangles

One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle?

Let the acute angles be A and B, with mA = 2x°.

mA + mB = 90°

2x+ mB = 90

Substitute 2x for mA.

mB = (90 – 2x)°

Subtract 2x from both sides.

4.1 Triangles and Angles

Acute s of rt. are comp.

Check It Out! Example 2a

The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle?

Let the acute angles be A and B, with mA = 63.7°.

mA + mB = 90°

63.7 + mB = 90

Substitute 63.7 for mA.

mB = 26.3°

Subtract 63.7 from both sides.

4.1 Triangles and Angles

Acute s of rt. are comp.

Check It Out! Example 2b

The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle?

Let the acute angles be A and B, with mA = x°.

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 = 48 .

Acute s of rt. are comp.

48 + mB = 90

Substitute 48 for mA.

mB = 41

Subtract 48 from both sides.

3° 5

2° 5

2° 5

2 5

2 5

2 5

Check It Out! Example 2c

The measure of one of the acute angles in a right triangle is 48 . What is the measure of the other acute angle?

mA + mB = 90°

The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure.

Exterior

Interior

4.1 Triangles and Angles

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.

4 is an exterior angle.

Exterior

Interior

3 is an interior angle.

4.1 Triangles and Angles

Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.

4 is an exterior angle.

The remote interior angles of 4 are 1 and 2.

Exterior

Interior

3 is an interior angle.

4.1 Triangles and Angles

4.1 Triangles and Angles

Example 3: Applying the Exterior Angle Theorem

Find mB.

mA + mB = mBCD

Ext. Thm.

Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.

15 + 2x + 3= 5x – 60

2x + 18= 5x – 60

Simplify.

Subtract 2x and add 60 to both sides.

78 = 3x

26 = x

Divide by 3.

mB = 2x + 3 = 2(26) + 3 = 55°

Check It Out! Example 3

Find mACD.

mACD = mA + mB

Ext. Thm.

Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB.

6z – 9 = 2z + 1+ 90

6z – 9= 2z + 91

Simplify.

Subtract 2z and add 9 to both sides.

4z = 100

z = 25

Divide by 4.

mACD = 6z – 9 = 6(25) – 9 = 141°

4.1 Triangles and Angles

From the figure, . So HF = 10, and EHF is isosceles.

Example 2A: Classifying Triangles by Side Lengths

Classify EHF by its side lengths.

4.1 Triangles and Angles

By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHG is scalene.

Example 2B: Classifying Triangles by Side Lengths

Classify EHGby its side lengths.

4.1 Triangles and Angles

From the figure, . So AC = 15, and ACD is isosceles.

Check It Out! Example 2

Classify ACD by its side lengths.

4.1 Triangles and Angles

Example 3: Using Triangle Classification

Find the side lengths of JKL.

Step 1 Find the value of x.

Given.

JK = KL

Def. of segs.

Substitute (4x – 10.7) for JK and (2x + 6.3) for KL.

4x – 10.7 = 2x + 6.3

Add 10.7 and subtract 2x from both sides.

2x = 17.0

x = 8.5

Divide both sides by 2.

4.1 Triangles and Angles

Example 3 Continued

Find the side lengths of JKL.

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

4.1 Triangles and Angles

Check It Out! Example 3

Find the side lengths of equilateral FGH.

Step 1 Find the value of y.

Given.

FG = GH = FH

Def. of segs.

Substitute

(3y – 4) for FG and (2y + 3) for GH.

3y – 4 = 2y + 3

Add 4 and subtract 2y from both sides.

y = 7

4.1 Triangles and Angles

Check It Out! Example 3 Continued

Find the side lengths of equilateral FGH.

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

4.1 Triangles and Angles

Example 4: Application

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?

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

4.1 Triangles and Angles

420 54 = 7 triangles

7 9

Example 4: Application Continued

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?

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.

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.

4.1 Triangles and Angles

Check It Out! Example 4a

Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.

The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.

P = 3(7)

P = 21 in.

4.1 Triangles and Angles

100 7 = 14 triangles

2 7

Check It Out! Example 4a Continued

Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel.

To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.

There is not enough steel to complete a fifteenth triangle. So the manufacturer can make 14 triangles from a 100 in. piece of steel.

4.1 Triangles and Angles

Check It Out! Example 4b

Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.

The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle.

P = 3(10)

P = 30 in.

4.1 Triangles and Angles

Check It Out! Example 4b Continued

Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel.

To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle.

100 10 = 10 triangles

The manufacturer can make 10 triangles from a 100 in. piece of steel.

4.1 Triangles and Angles

Lesson Quiz I

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

33 °

4.1 Triangles and Angles

2 3

Lesson Quiz: Part II

5. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle?

6. Find mABD.

124°

4.1 Triangles and Angles

Lesson Quiz: Part III

7. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store?

30°

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