Digits, Grade 7, Volume 1, Homework Helper

#### Part 1

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.

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