Question

Quadrilateral RSTU, diagonals SU and RT intersect at point V. RSTU is a parallelogram. If m∠TSV = 31° and m∠SVT = 126°, explain how you can find the measure of ∠URV. Show all steps of your work, and refer to any properties of triangles, parallelograms, or triangle congruency theorems as necessary to justify your response.

Answer 1

The measure of angle ∠URV is 49°. This can be determined by using the properties of parallelograms and the Triangle Angle Sum Theorem, along with the given angle measures.

Step-by-step explanation:

In a parallelogram, opposite angles are equal. Given that RSTU is a parallelogram, angles ∠TSV and ∠SVT are opposite angles and therefore have the same measure. So, m∠TSV = m∠SVT = 31°. Additionally, the opposite angles of a parallelogram sum to 180°. Therefore, m∠RUS + m∠TSV = 180°. Solving for m∠RUS, we have m∠RUS = 180° - 31° = 149°.

Now, considering the diagonal SU, ∠URV is an exterior angle to triangle RUS. According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Therefore, m∠URV = m∠RUS + m∠TSV. Substituting the known values, m∠URV = 149° + 31° = 180°. However, this is a straight angle, and we need to find the actual angle measure, so m∠URV = 180° - m∠SVT. Substituting m∠SVT = 126°, we get m∠URV = 180° - 126° = 54°. Therefore, the measure of ∠URV is 54°.

In summary, using the properties of parallelograms and the Triangle Angle Sum Theorem, along with the Exterior Angle Theorem, we can determine that the measure of ∠URV in parallelogram RSTU is 54°.

Answer 2

The measure of ∠URV is 23°.

To find the measure of ∠URV, we can use the properties of parallelograms and triangle congruence theorems.

1. Since RSTU is a parallelogram, opposite angles are congruent. Therefore, ∠R = ∠U and ∠S = ∠T.

2. Since m∠TSV = 31° and m∠SVT = 126°, we can find m∠STV by subtracting these angles from 180° (the sum of the angles in a triangle). Thus, m∠STV = 180° - 31° - 126° = 23°.

3. Now, using the fact that ∠S = ∠T, we can conclude that m∠TSV = m∠SVT = 23°.

4. Since the sum of the angles in a triangle is 180°, we can find m∠TVS (which is equal to ∠U) by subtracting the known angles from 180°: m∠TVS = 180° - 23° - 23° = 134°.

5. Finally, using the fact that opposite angles in a parallelogram are congruent, we can conclude that ∠URV is congruent to ∠STV. Therefore, m∠URV = m∠STV = 23°.