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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles

3-5 Parallel Lines and Triangles

Teks Focus TEKS (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas.

Additional TEKS (1)(G)

Vocabulary • Auxiliary line – a line that you add to a diagram to help explain relationships in proofs

• Exterior angle of a polygon – an angle formed by a side and an extension of an adjacent side

• Remote interior angles – the two nonadjacent interior angles corresponding to each exterior angle of a triangle

• Analyze – closely examine objects, ideas, or relationships to learn more about their nature

ESSENTIAL UNDERSTANDING

The sum of the angle measures of a triangle is always the same.

take note Postulate 3-2 Parallel Postulate

Through a point not on a line, there is one and only one line parallel to the given line.

There is exactly one line through P parallel to ℓ.

take note Theorem 3-11 Triangle Angle-Sum Theorem

The sum of the measures of the angles of a triangle is 180.

m∠A + m∠B + m∠C = 180 For a proof of Theorem 3-11, see Problem 1.

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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles

take note Key Concept Angles of Polygons

An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side. For each exterior angle of a triangle, the two nonadjacent interior angles are its remote interior angles. In each triangle below, ∠1 is an exterior angle and ∠2 and ∠3 are its remote interior angles.

Theorem 3-12 Triangle Exterior Angle Theorem

The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

m∠1 = m∠2 + m∠3

You will prove Theorem 3-12 in Exercise 20.

Problem 1

TEKS Process Standard (1)(G)

Proof Proving the Triangle Angle-Sum Theorem

Write a two-column proof to prove the Triangle Angle-Sum Theorem (Theorem 3-11). Plan What could you add to the diagram to help explain relationships in this proof?

Add an auxiliary line through Z, ¯¯¯¯¯¯ parallel to XY .

Given: ΔAXYZ

Prove: m∠X + m∠2 + m∠Y = 180

Statements Reasons ← → ¯¯¯¯¯¯ 1) Draw AB through Z, parallel to XY . 1) Parallel Postulate 2) ∠1 and ∠XZB are supplementary. 2) Linear Pair Postulate 3) m∠1 + m∠XZB = 180 3) Definition of supplementary angles 4) m∠XZB = m∠2 + m∠3 4) Angle Addition Postulate 5) m∠1 + m∠2 + m∠3 = 180 5) Substitution Property 6) ∠1 ≅∠X and ∠3 ≅∠Y 6) If lines are parallel, then alternate interior angles are congruent. 7) m∠1 = m∠X and m∠3 = m∠Y 7) Definition of congruent angles 8) m∠X + m∠2 + m∠Y = 180 8) Substitution Property

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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles

Problem 2

TEKS Process Standard (1)(F)

Using the Triangle Angle-Sum Theorem

Algebra What are the values of x and y in the diagram at the right? Plan Which variable should you solve for first? d From the diagram, you know two angle measures in ΔADB. The third angle is labeled x°. So Think Write use what you know about the Use the Triangle Angle-Sum Theorem to write an equation involving x. 59 + 43 + x = 180 angle measures in a triangle to solve for x first. 102 + x = 180 Solve for x by simplifying and then subtracting 102 from each side. x = 78 ∠ADB and ∠CDB form a linear pair, so they are supplementary. m∠ADB + m∠CDB = 180 x + y = 180 Substitute 78 for m ADB and y for m CDB in the above equation. ∠ ∠ 78 + y = 180 Solve for y by subtracting 78 from each side. y = 102

Problem 3

Using the Triangle Exterior Angle Theorem

A What is the measure of ∠1? Plan m 1 = 80 + 18 Triangle Exterior Angle Theorem ∠ What information can you get m∠1 = 98 Simplify. from the diagram?

The diagram shows you which angles are interior or exterior.

B What is the measure of ∠2?

124 = 59 + m∠2 Triangle Exterior Angle Theorem 65 = m∠2 Subtract 59 from each side.

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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles

Problem 4

Applying the Triangle Theorems

Multiple Choice When radar tracks an object, the reflection of signals off the ground can result in clutter. Clutter causes the receiver to confuse the real object with its reflection, called a ghost. At the right, there is a radar receiver at A, an airplane at B, and the airplane's ghost at D. What is the value of x? Plan How can you apply your A 30 skills from Problem 3 here?

B 50 Look at the diagram. Notice that you have a triangle and C 70 information about interior and exterior angles. D 80

m∠A + m∠B = m∠BCD Triangle Exterior Angle Theorem x + 30 = 80 Substitute. x = 50 Subtract 30 from each side. The value of x is 50. The correct answer is B.

PRACTICE and APPLICATION EXERCISES

Proof 1. Justify Mathematical Arguments (1)(G) Write a paragraph proof to prove the Triangle Angle-Sum Theorem (Theorem 3-11). Begin by drawing an auxiliary line through vertex T. Scan page for a Virtual Given: ΔSTU Nerd™ tutorial video.

Prove: m∠S + m∠T + m∠U = 180

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support when completing your Find the value of each variable. homework, go to PearsonTEXAS.com .

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Find each missing angle measure.

6. Expand this image

7. Expand this image

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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles

8. A ramp forms the angles shown at the right. What are the values of a and b?

9. Analyze Mathematical Relationships (1)(F) What is the measure of each angle of a triangle with three congruent angles? Explain.

10. A beach chair has different settings that change the angles formed by its parts. Suppose m∠2 = 71 and m∠3 = 43. Find m∠1.

Use the given information to find the unknown angle measures in the triangle.

11. The ratio of the angle measures of the acute angles in a right triangle is 1 : 2.

12. The measure of one angle of a triangle is 40. The measures of the other two angles are in a ratio of 3 : 4.

13. The measure of one angle of a triangle is 108. The measures of the other two angles are in a ratio of 1 : 5.

14. Analyze Mathematical Relationships (1)(F) The angle measures of ΔRST are represented by 2x, x + 14, and x − 38. What are the angle measures of ΔRST?

Proof 15. Prove the following theorem: The acute angles of a right triangle are complementary.

Given: ΔABC with right angle C

Prove: ∠A and ∠B are complementary.

Find the values of the variables and the measures of the angles.

16.

17.

18.

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19. Expand this image

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Unit 1 Logical Arguments and Constructions; Proof and Congruence > Topic 3 Parallel and Perpendicular Lines > 3-5 Parallel Lines and Triangles

Proof 20. Prove the Triangle Exterior Angle Theorem (Theorem 3-12). The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

Given: ∠1 is an exterior angle of the triangle.

Prove: m∠1 = m∠2 + m∠3 21. Without using the Triangle Angle-Sum Theorem as a reason, write a two-column proof to prove that the acute angles of a right triangle are complementary.

Given: ΔABC with right angle ACB

Prove: ∠BAC and ∠ABC are complementary. 22. Explain Mathematical Ideas (1)(G) Two angles of a triangle measure 64 and 48. What is the measure of the largest exterior angle of the triangle? Explain.

23. Analyze Mathematical Relationships (1)(F) A right triangle has exterior angles at each of its acute angles with measures in the ratio 13 : 14. Find the measures of the two acute angles of the right triangle. ¯¯¯¯¯ ¯¯¯¯¯ ¯¯¯¯¯ 24. In the figure at the right, CD ⊥ AB and CD bisects ∠ACB. Find m∠DBF. 25. If the remote interior angles of an exterior angle of a triangle are congruent, what can you conclude about the bisector of the exterior angle? Justify your answer.

TEXAS Test Practice

26. The measure of one angle of a triangle is 115. The other two angles are congruent. What is the measure of each of the congruent angles?

A. 32.5

B. 57.5

C. 65

D. 115

27. One statement in a proof is “∠1 and ∠2 are supplementary angles.” The next statement is “m∠1 + m∠2 = 180.” Which is the best justification for the second statement based on the first statement?

F. The sum of the measures of two right angles is 180.

G. Angles that form a linear pair are supplementary.

H. Definition of supplementary angles

J. The measure of a straight angle is 180.

28. ΔABC has one obtuse angle, m∠A = 21, and ∠C is acute.

a. What is m∠B + m∠C? Explain.

b. What is the range of whole numbers for m∠C? Explain.

c. What is the range of whole numbers for m∠B? Explain.

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