Section 4 - Proofs with Angles and Transversals

Angles have many theorems that can be used to prove new ideas.

Definitions-

angles a and b are a linear pair. a + b = 180 degrees

angles c and d are vertical angles. c = d

Theorems-

-If two angles are a linear pair, then they are supplementary

-Angles that are supplementary to the same angle or to congruent angles are congruent

-Angles that are complementary to the same angle or to congruent angles are congruent

-All right angles are congruent

-Vertical angles are congruent

-Perpendicualr lines intersect to form 4 right angles

-a + b = c . (angle addition theorem.

A transversal is a line that crosses two parallel lines.

Line m is parallel to n, so p is the transversal. All of the new postulates for transversals involve the angles formed. The angles formed by the transversal are named for their relation to the transversal.

angles d,e,a and h are exterior angles. angles c,f,b and g are interior angles

-opposite angles in a transversal are congruent a = g and c = e

-adjacent angles are supplementary, and they add up to 180 degrees. h + g = 180 degrees, e + f = 180 degrees.

-alternate interior angles are congruent, b = f and c = g

-consecutive interior angles are supplementary, b + c = 180, g + f = 180

-alternate exterior angles are congruent, a = e, d = h

-consecutive exterior angles are supplementary, a + d = 180, h + e = 180

-corresponding angles are congruent, g = e, b = d, c = a, f = h

The proofs for angles and transversals are still written in the two column form, you just have more postulates now.

Fill in the blank with always, sometimes, or never.

1) Vertical angles are ______ congruent.

2) Congruent angles are _______ right angles.

3) Complementary angles are ______ congruent.

4) Transversals are ______ perpendicular to the lines they intersect.

5) Adjacent angles in a transversal are _______ supplementary.

Find the value of x -

6) 7)

8) 9)

10) Write a two-column proof-

Given: angles 1 and 2 form a linear pair

Prove: angles 1 and 2 are supplementary

11) Given: angle ABC = angle EFG , angle ABD = angle EFH

Prove: angle DBC = angle HFG

Find the measure of each angle. Questions 12 - 18

12) 3

13) 6

14) 2

15) 7

16) 5

17) 4

18) 1

19) Given: p || q

Prove: angles 1 and 2 are supplementary

20) Given m || n

Prove: angles 1 and 4 are supplementary