3-5 Parallel Lines and Triangles

Quick Review

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

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

Example

What are the values of x and y?

x + 50 = 125 Exterior Angle Theorem x = 75 Simplify . table with 2 rows and 2 columns , row1 column 1 , x plus 50 equals 125 , column 2 cap exteriorcap anglecap theorem , row2 column 1 , x equals 75 , column 2 cap simplify , . , end table

x + y + 50 = 180 Triangle Angle-Sum Theorem 75 + y + 50 = 180 Substitute  75  for  x . y = 55 Simplify . table with 3 rows and 3 columns , row1 column 1 , x plus y plus 50 , column 2 equals 180 , column 3 cap trianglecap angleminuscap sumcap theorem , row2 column 1 , 75 plus y plus 50 , column 2 equals 180 , column 3 cap substitute . 75 , for , x . , row3 column 1 , y , column 2 equals 55 , column 3 cap simplify , . , end table

Exercises

Find the values of the variables.

The measures of the three angles of a triangle are given. Find the value of x.

27. x, 2x, 3x
28. x + 10 , x − 20 , x + 25 x plus 10 comma x . minus 20 comma , x plus 25
29. 20 x + 10 , 30 x − 2 , 7 x + 1 20 x plus 10 comma . 30 x minus 2 comma , 7 x plus 1

3-6 Constructing Parallel and Perpendicular Lines

Quick Review

You can use a compass and a straightedge to construct

• a line parallel to a given line through a point not on the line
• a line perpendicular to a given line through a point on the line, or through a point not on the line

Example

Which step of the parallel lines construction guarantees the lines are parallel?

Image Long Description

The parallel lines construction involves constructing a pair of congruent angles. Since the congruent angles are corresponding angles, the lines are parallel.

Exercises

30. Draw a line m and point Q not on m. Construct a line perpendicular to m through Q.

Use the segments below.

31. Construct a rectangle with side lengths a and b.
32. Construct a rectangle with side lengths a and 2b.
33. Construct a quadrilateral with one pair of parallel opposite sides, each side of length 2a.

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