Question

practice! justify each statement below using a property of equality, property of congruence, definition, postulate, or theorem. 1. if ∠c is a right angle, then m∠c = 90°. 2. if ∠x is supplementary to ∠y and ∠x is supplementary to ∠z, then ∠y≅∠z. 3. if image of intersecting lines with angles 1 and 2 then, ∠1≅∠2. 4. if m∠p + m∠q = 90°, then ∠p and ∠q are complementary. 5. if ∠m and ∠n form a right angle, then ∠m and ∠n are complementary. 6. given: image of perpendicular lines l and m with angle 1 if l⊥m, then ∠1 is a right angle. 7. if ∠w and ∠x are supplementary, then m∠w + m∠x = 180°. 8. if ∠l is complementary to ∠m and ∠n is complementary to ∠m, then ∠l≅∠n. 9. if ∠a and ∠b form a linear pair, then ∠a and ∠b are supplementary. 10. if ∠n and ∠p are complementary, then m∠n + m∠p = 90°. 11. given the diagram to the right: image of angle jkl with point m in the interior m∠jkm + m∠mkl = m∠jkl. 12. if m∠r = m∠s, then ∠r≅∠s. reasons bank properties of equality: addition property, subtraction property, multiplication property, division property, distributive property, substitution property, reflexive property, symmetric property, transitive property properties of congruence: reflexive property, symmetric property, transitive property postulates: angle addition postulate theorems: vertical angles theorem, complement theorem, linear pair (supplement) theorem, congruent complements theorem, congruent supplements theorem definitions: definition of congruence, definition of a right angle, definition of complementary angles, definition of supplementary angles, definition of an angle bisector, definition of perpendicular

Explanation:

Step1: Recall right - angle definition

A right angle is defined as an angle with a measure of 90 degrees. So, if $\angle C$ is a right angle, then $m\angle C = 90^{\circ}$ by the definition of a right angle.

Step2: Analyze supplementary angles

If $\angle X$ is supplementary to $\angle Y$ ($m\angle X+m\angle Y = 180^{\circ}$) and $\angle X$ is supplementary to $\angle Z$ ($m\angle X + m\angle Z=180^{\circ}$), then by the transitive property of equality, $m\angle Y=m\angle Z$, and $\angle Y\cong\angle Z$ by the definition of congruence.

Step3: Identify vertical angles

The angles $\angle1$ and $\angle2$ are vertical angles. By the vertical angles theorem, vertical angles are congruent, so $\angle1\cong\angle2$.

Step4: Apply complementary - angle definition

If $m\angle P + m\angle Q=90^{\circ}$, then by the definition of complementary angles, $\angle P$ and $\angle Q$ are complementary.

Step5: Consider angle - sum in a right - angle

If $\angle M$ and $\angle N$ form a right angle, then $m\angle M + m\angle N = 90^{\circ}$, and by the definition of complementary angles, $\angle M$ and $\angle N$ are complementary.

Step6: Use perpendicular - line definition

If $l\perp m$, by the definition of perpendicular, the angles formed are right angles. So, $\angle1$ is a right angle.

Step7: Apply supplementary - angle definition

If $\angle W$ and $\angle X$ are supplementary, then by the definition of supplementary angles, $m\angle W + m\angle X=180^{\circ}$.

Step8: Use congruent - complements theorem

If $\angle L$ is complementary to $\angle M$ and $\angle N$ is complementary to $\angle M$, then by the congruent - complements theorem, $\angle L\cong\angle N$.

Step9: Recall linear - pair theorem

If $\angle A$ and $\angle B$ form a linear pair, then by the linear - pair (supplement) theorem, $\angle A$ and $\angle B$ are supplementary.

Step10: Apply complementary - angle definition

If $\angle N$ and $\angle P$ are complementary, then by the definition of complementary angles, $m\angle N + m\angle P=90^{\circ}$.

Step11: Use angle - addition postulate

For the angles $\angle JKM$, $\angle MKL$ and $\angle JKL$, by the angle - addition postulate, $m\angle JKM + m\angle MKL=m\angle JKL$.

Step12: Apply definition of congruence

If $m\angle R=m\angle S$, then by the definition of congruence, $\angle R\cong\angle S$.

Answer:

1. Definition of a right angle
2. Transitive property of equality and definition of congruence
3. Vertical angles theorem
4. Definition of complementary angles
5. Definition of complementary angles
6. Definition of perpendicular
7. Definition of supplementary angles
8. Congruent - complements theorem
9. Linear - pair (supplement) theorem
10. Definition of complementary angles
11. Angle - addition postulate
12. Definition of congruence