Example Making Inferences About Vertical and Supplementary Angles
Determine whether each statement about the diagram is true or false.
a. Eight angles each measure 180° or less.
b. If ∠ 4 is acute, then ∠ 1 is obtuse.
c. If m ∠ 2 = 90°, then m ∠ 3 = m ∠ 4.
d. m ∠ 1 + m ∠ 3 = m ∠ 2 + m ∠ 4
e. (m ∠ 1 + m ∠ 2) − (m ∠ 3 + m ∠ 4) = 0
Solution
a. Angles that measure less than 180°: ∠ 1, ∠ 2, ∠ 3, ∠ 4
Angles that measure 180°: ∠ 1 + ∠ 2, ∠ 1 + ∠ 3, ∠ 2 + ∠ 4, ∠ 3 + ∠ 4
There are eight angles that measure 180° or less.
The statement is true.
b. ∠ 4 and ∠ 1 are vertical angles, so their measure are equal. So if ∠ 4 is acute, then ∠ 1 must also be acute.
The statement is false.
c. ∠ 3 and ∠ 4 are supplementary angles, so the sum of their measures is 180°. ∠ 2 and ∠ 3 are vertical angles, so their measures are equal. So m ∠ 3 = 90°.
table with 4 rows and 2 columns , row1 column 1 , m angle 3 plus m angle 4 , column 2 equals 180 degrees , row2 column 1 , 90 degrees plus m angle 4 , column 2 equals 180 degrees , row3 column 1 , 90 degrees plus m angle 4 minus 90 degrees , column 2 equals 180 degrees negative 90 degrees , row4 column 1 , m angle 4 , column 2 equals 90 degrees , end table
The statement is true.
d. ∠ 1 and ∠ 3 are supplementary angles, so the sum of their measures is 180°.
∠ 2 and ∠ 4 are supplementary angles, so the sum of their measure is also 180°.
The statement is true.
e. ∠ 1 and ∠ 2 are supplementary angles, so the sum of their measures is 180°. ∠ 3 and ∠ 4 are supplementary angles, so the sum of their measures is also 180°.
table with 3 rows and 2 columns , row1 column 1 , open m angle 1 plus m angle 2 close minus open m angle 3 plus m angle 4 close , column 2 equals 0 , row2 column 1 , 180 degrees negative 180 degrees , column 2 equals 0 , row3 column 1 , 0 , column 2 equals 0 , end table
The statement is true.