Question

u1t3d5: angle pairs – the big problem! front side for points. back page for extra practice review: what angle relationships do we know so far? list all angle relationships that result in congruent angles when on parallel lines cut by a transversal: list all angle relationships that result in supplementary angles when on parallel lines cut by a transversal: find the measure of each angle in the diagram below. name any relationship you use to help you find each measure. (reason example: alternate interior angle with ∠e or ∠110). m∠a = ____ reason:__ m∠c = __ reason:__ m∠d = __ reason:__ m∠e = __ reason:__ m∠f = __ reason:__ m∠h = __ reason:__ m∠i = __ reason:__ m∠j = __ reason:____

Explanation:

Step1: Recall congruent - angle relationships

When parallel lines are cut by a transversal, the congruent - angle relationships are: corresponding angles, alternate interior angles, and alternate exterior angles.

Step2: Recall supplementary - angle relationships

When parallel lines are cut by a transversal, the supplementary - angle relationships are: same - side interior angles and same - side exterior angles.

Step3: Find \(m\angle a\)

\(\angle a\) and the \(150^{\circ}\) angle are supplementary (linear pair). So \(m\angle a=180 - 150=30^{\circ}\). Reason: linear - pair angles are supplementary.

Step4: Find \(m\angle c\)

\(\angle c\) and the \(140^{\circ}\) angle are supplementary (linear pair). So \(m\angle c = 180-140 = 40^{\circ}\). Reason: linear - pair angles are supplementary.

Step5: Find \(m\angle d\)

\(\angle d\) and \(\angle c\) are alternate interior angles. Since the lines are parallel, \(m\angle d=m\angle c = 40^{\circ}\). Reason: alternate interior angles are congruent.

Step6: Find \(m\angle e\)

\(\angle e\) and the \(110^{\circ}\) angle are alternate exterior angles. Since the lines are parallel, \(m\angle e = 110^{\circ}\). Reason: alternate exterior angles are congruent.

Step7: Find \(m\angle f\)

\(\angle f\) and the \(110^{\circ}\) angle are supplementary (same - side interior angles). So \(m\angle f=180 - 110 = 70^{\circ}\). Reason: same - side interior angles are supplementary.

Step8: Find \(m\angle h\)

\(\angle h\) and \(\angle f\) are alternate interior angles. Since the lines are parallel, \(m\angle h=m\angle f = 70^{\circ}\). Reason: alternate interior angles are congruent.

Step9: Find \(m\angle i\)

\(m\angle i = 110^{\circ}\) because it is vertical to the \(110^{\circ}\) angle. Reason: vertical angles are congruent.

Step10: Find \(m\angle j\)

\(\angle j\) and \(\angle i\) are supplementary (linear pair). So \(m\angle j=180 - 110 = 70^{\circ}\). Reason: linear - pair angles are supplementary.

Answer:

\(m\angle a = 30^{\circ}\), reason: linear - pair angles are supplementary
\(m\angle c = 40^{\circ}\), reason: linear - pair angles are supplementary
\(m\angle d = 40^{\circ}\), reason: alternate interior angles are congruent
\(m\angle e = 110^{\circ}\), reason: alternate exterior angles are congruent
\(m\angle f = 70^{\circ}\), reason: same - side interior angles are supplementary
\(m\angle h = 70^{\circ}\), reason: alternate interior angles are congruent
\(m\angle i = 110^{\circ}\), reason: vertical angles are congruent
\(m\angle j = 70^{\circ}\), reason: linear - pair angles are supplementary