Question

match the theorems below.
if ∠j and ∠k are vertical angles, then ∠j = ∠k
if ∠o and ∠w form a right angle, then ∠o and ∠w are complementary angles
if ∠abc + ∠xyz = 180°, then ∠abc and ∠xyz are supplementary angles
if ∠l is supplementary to ∠m and ∠j is supplementary to ∠m, then ∠j = ∠l
if ∠abd = 90°, then ∠abd is a right angle
if b is in the interior of ∠jkl, then ∠jkb + ∠bkl = ∠jkl
if ∠1 and ∠2 are complementary, then m∠1 + m∠2 = 90°
if ∠a and ∠z form a linear pair, then ∠a and ∠z are supplementary angles
if m∠3 = m∠4, then ∠3 = ∠4
if ∠2 is complementary to ∠4 and ∠5 is complementary to ∠4, then ∠2 = ∠5

Explanation:

• If \( \angle J\) and \( \angle K\) are vertical angles, then \( \angle J=\angle K\): This is the Vertical Angles Theorem which states that vertical angles (angles opposite each other when two lines intersect) are congruent.
• If \( \angle O\) and \( \angle W\) form a right angle, then \( \angle O\) and \( \angle W\) are complementary angles: By the Definition of Complementary Angles (two angles whose sum is \(90^{\circ}\), and a right - angle is \(90^{\circ}\)).
• If \( \angle ABC+\angle XYZ = 180^{\circ}\), then \( \angle ABC\) and \( \angle XYZ\) are supplementary angles: According to the Definition of Supplementary Angles (two angles whose sum is \(180^{\circ}\)).
• If \( \angle L\) is supplementary to \( \angle M\) and \( \angle J\) is supplementary to \( \angle M\), then \( \angle J=\angle L\): This is the Congruent Supplements Theorem (if two angles are supplementary to the same angle, then they are congruent).
• If \( \angle ABD = 90^{\circ}\), then \( \angle ABD\) is a right angle: By the Definition of a Right Angle (\(90^{\circ}\) angle is a right angle).
• If \(B\) is in the interior of \( \angle JKL\), then \( \angle JKB+\angle BKL=\angle JKL\): This is the Angle Addition Postulate (if a point lies in the interior of an angle, the sum of the two smaller angles formed is equal to the larger angle).
• If \( \angle 1\) and \( \angle 2\) are complementary, then \(m\angle1 + m\angle2=90^{\circ}\): By the Definition of Complementary Angles (the sum of measures of complementary angles is \(90^{\circ}\)).
• If \( \angle A\) and \( \angle Z\) form a linear pair, then \( \angle A\) and \( \angle Z\) are supplementary angles: This is the Supplements Theorem (a linear pair of angles is supplementary).
• If \(m\angle3=m\angle4\), then \( \angle3=\angle4\): By the Definition of Congruence (if the measures of two angles are equal, the angles are congruent).
• If \( \angle 2\) is complementary to \( \angle 4\) and \( \angle 5\) is complementary to \( \angle 4\), then \( \angle 2=\angle 5\): This is the Congruent Complements Theorem (if two angles are complementary to the same angle, then they are congruent).

Answer:

• If \( \angle J\) and \( \angle K\) are vertical angles, then \( \angle J=\angle K\): Vertical Angles Theorem
• If \( \angle O\) and \( \angle W\) form a right angle, then \( \angle O\) and \( \angle W\) are complementary angles: Definition of Complementary Angles
• If \( \angle ABC+\angle XYZ = 180^{\circ}\), then \( \angle ABC\) and \( \angle XYZ\) are supplementary angles: Definition of Supplementary Angles
• If \( \angle L\) is supplementary to \( \angle M\) and \( \angle J\) is supplementary to \( \angle M\), then \( \angle J=\angle L\): Congruent Supplements Theorem
• If \( \angle ABD = 90^{\circ}\), then \( \angle ABD\) is a right angle: Definition of a Right Angle
• If \(B\) is in the interior of \( \angle JKL\), then \( \angle JKB+\angle BKL=\angle JKL\): Angle Addition Postulate
• If \( \angle 1\) and \( \angle 2\) are complementary, then \(m\angle1 + m\angle2=90^{\circ}\): Definition of Complementary Angles
• If \( \angle A\) and \( \angle Z\) form a linear pair, then \( \angle A\) and \( \angle Z\) are supplementary angles: Supplements Theorem
• If \(m\angle3=m\angle4\), then \( \angle3=\angle4\): Definition of Congruence
• If \( \angle 2\) is complementary to \( \angle 4\) and \( \angle 5\) is complementary to \( \angle 4\), then \( \angle 2=\angle 5\): Congruent Complements Theorem