Angles of Triangles

LESSON 4–2

Five-Minute Check (over Lesson 4–1)

TEKS

Then/Now

New Vocabulary

Theorem 4.1: Triangle Angle-Sum Theorem

Proof: Triangle Angle-Sum Theorem

Example 1: Real-World Example: Use the Triangle Angle-Sum Theorem

Theorem 4.2: Exterior Angle Theorem

Proof: Exterior Angle Theorem

Example 2: Real-World Example: Use the Exterior Angle Theorem

Corollaries: Triangle Angle-Sum Corollaries

Example 3: Find Angle Measures in Right Triangles

Over Lesson 4–1

A. acute

B. equiangular

C. obtuse

D. right

Classify ΔRST .

Over Lesson 4–1

A. 8

B. 10

C. 12

D. 14

Find y if ΔRST is an isosceles triangle with RS RT.___ ___

Over Lesson 4–1

A. 2

B. 4

C. 6

D. 8

Find x if ΔABC is an equilateral triangle.

Over Lesson 4–1

A. ΔABC

B. ΔACB

C. ΔADC

D. ΔCAB

Over Lesson 4–1

A. scalene

B. isosceles

C. equilateral

Classify ΔMNO as scalene, isosceles, or equilateral if MN = 12, NO = 9, and MO = 15.

Over Lesson 4–1

A. acute

B. scalene

C. isosceles

D. equiangular

Which is not a classification for ΔFGH?

Targeted TEKSG.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum ofinterior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships tosolve problems.

Mathematical ProcessesG.1(D), G.1(F)

You classified triangles by their side or angle measures.

• Apply the Triangle Angle-Sum Theorem.

• Apply the Exterior Angle Theorem.

• auxiliary line

• exterior angle

• remote interior angles

• flow proof

• corollary

Use the Triangle Angle-Sum Theorem

SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle.

Analyze Examine the information in the diagram. You know the measures of two angles of one triangle and only one measure of another. You also know that 1 and 2 are vertical angles.

Use the Triangle Angle-Sum Theorem

Triangle Angle-Sum Theorem

Simplify.

Subtract 117 from each side.

Formulate Find m1 first because the measure of two angles of the triangle are known. Use the Vertical Angles Theorem to find m2. Then you will have enough information to find the measure of 3.

Determine

Use the Triangle Angle-Sum Theorem

Triangle Angle-Sum Theorem

Simplify.

Subtract 142 from each side.

1 and 2 are congruent vertical angles. So, m2 = 63.

Answer: Therefore, m1 = 63, m2 = 63, and m3 = 38.

Justify The sums of the measures of the angles in each triangle should be 180.

m1 + 43 + 74 = 63 + 43 + 74 or 180m2 + m3 + 79= 63 + 38 + 79 or 180

Use the Triangle Angle-Sum Theorem

Evaluate By identifying each part of the problem, this complex problem could be separated into three manageable pieces. The properties of triangles wereused to check the reasonableness of the answers found.

A. 95

B. 75

C. 57

D. 85

Find the measure of 3.

Use the Exterior Angle Theorem

GARDENING Find the measure of FLW in the fenced flower garden shown.

mLOW + mOWL = mFLW Exterior Angle Theorem

x + 32 = 2x – 48 Substitution

32 = x – 48 Subtract x from each side.

80 = x Add 48 to each side.

Answer: So, mFLW = 2(80) – 48 or 112.

A. 30

B. 40

C. 50

D. 130

The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD.

Find Angle Measures in Right Triangles

Find the measure of each numbered angle.

If 2 s form a linear pair, they are supplementary.

Exterior Angle Theoremm1 = 38 + 32

Simplify.= 70

Substitution70 + m2 = 180

Subtract 70 from each side.110

Find Angle Measures in Right Triangles

Subtract 78 from each side.

102

Simplify.78+ m4 = 180

If 3 s form a linear pair, they are supplementary

46+ 32+ m 4 = 180

Simplify.= 46

m 3 + 64 = 110 Exterior Angle Theorem

Find Angle Measures in Right Triangles

Subtract 143 from each side.37

Simplify.m5 + 143 = 180

Triangle Angle-Sum Theoremm5 + 102+ 41 = 180

m1 = 70, m2 = 110, m3 = 46, m4 = 102, m5 =37

A. 50

B. 45

C. 85

D. 130

Find m3.

Angles of Triangles

LESSON 4–2