Digits, Grade 7, Volume 1, Homework Helper

3-7: Additional Problem Solving

Part 1

Example Making Inferences About Vertical and Supplementary Angles

Determine whether each statement about the diagram is true or false.

A diagram with two intersecting lines, creating four angles, labeled counterclockwise 1, 2, 4, 3.

  • 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.


End ofPage 118

Table of Contents

Digits, Grade 7, Volume 1, Homework Helper Unit A: Number and Operations Unit B: Expressions, Equations, and Relationships Unit C: Measurement and Geometry Unit D: Proportionality English/Spanish Glossary Formulas Math Symbols Formulas Measures Properties