Question

1. \if $angle a$ and $angle b$ are complementary, then $mangle a + mangle b = 90^circ$\ a. complement theorem b. definition of a right angle c. congruent complements theorem d. definition of complementary angles 20. given: \if $l perp m$, then $angle 1$ is a right angle\ a. definition of a right angle b. definition of complementary angles c. definition of perpendicular d. complement theorem complete each proof. 21. given: $overline{wy} cong overline{xz}$ prove: $overline{wx} cong overline{yz}$ statements 1. $overline{wy} cong overline{xz}$ 2. $wy = xz$ 3. $wx + xz = wz\\ wy + yz = wz$ 4. $xz + yz = wz$ 5. $wx + xz = xz + yz$ 6. $wx = yz$ 7. $overline{wx} cong overline{yz}$ reasons 1. 2. 3. 4. 5. 6. 7. 22. given: $d$ is the midpoint of $overline{ce}$ prove: $de = \frac{1}{2}ce$ statements 1. $d$ is the midpoint of $overline{ce}$ 2. $cd = de$ 3. $cd + de = de + de$ 4. $cd + de = 2de$ 5. $cd + de = ce$ 6. $2de = ce$ 7. $de = \frac{1}{2}ce$ reasons 1. 2. 3. 4. 5. 6. 7. © gina wilson (all things algebra®, llc), 2014-2020

Explanation:

Questions 19-20 (Multiple Choice)

1. The statement defines the core property of complementary angles: their measures add to 90°.
2. The statement describes the defining characteristic of perpendicular lines: they form a right angle.

Step1: State given information

$\overline{WY} \cong \overline{XZ}$

Step2: Congruent segments have equal length

$WY = XZ$

Step3: Segment Addition Postulate (W-X-Y-Z)

$WX + XZ = WZ; WY + YZ = WZ$

Step4: Transitive Property of Equality

$WX + XZ = WY + YZ$

Step5: Substitute $WY=XZ$ (from Step2)

$WX + XZ = XZ + YZ$

Step6: Subtract $XZ$ from both sides

$WX = YZ$

Step7: Equal lengths mean congruent segments

$\overline{WX} \cong \overline{YZ}$

Step1: State given information

$D$ is the midpoint of $\overline{CE}$

Step2: Midpoint divides segment into 2 equal parts

$CD = DE$

Step3: Add $DE$ to both sides

$CD + DE = DE + DE$

Step4: Simplify right-hand side

$CD + DE = 2DE$

Step5: Segment Addition Postulate (C-D-E)

$CD + DE = CE$

Step6: Substitute $CD+DE=CE$ (from Step5)

$2DE = CE$

Step7: Divide both sides by 4

$DE = \frac{1}{4}CE$

Answer:

1. D. Definition of Complementary Angles
2. C. Definition of Perpendicular

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Question 21 (Proof Completion)