Question

In the given figure, if TU ∥ SR and TR ∥ SV, then find ∠a and ∠b.

Answer 1

Answer:

Measure of ∠a is 115° and ∠b is 40°.

Step-by-step explanation:

Given TU ∥ SR and TR ∥ SV

Since TU is extended to A, thus TA ∥ SR.

So, ATRS is a parallelogram. And in a parallelogram opposite angles are equal.

Given $\angle ATR = 90^o$ and hence $\angle ASR = 90^o$

And the other pair of opposite angles must be 180° and equal,

Therefore, $\angle TAS = 90^o$ and  $\angle TRS = 90^o$

$\angle TAS~\&~\angle TAV$ form a linear pair, hence the sum of the angles is 180°.

$\implies \angle TAS+\angle TAV=180^o$

$\implies 90^o+\angle TAV=180^o$

$\implies \angle TAV=180^o-90^o=90^o$

In a triangle, the sum of all the angles is 180°.

In triangle VAU, $\angle UAV+\angle AVU+\angle VUA=180^o$

$\implies 90^o+25^o+\angle VUA=180^o$

$\implies 115^o+\angle VUA=180^o$

$\implies \angle VUA=180^o-115^o=65^o$  

$\angle VUA~\&~\angle VUT$ form a linear pair, thus

$\angle VUA+\angle VUT=180^o$

$\implies 65^o+\angle VUT=180^o$

$\implies \angle VUT=180^o-65^o=115^o$

Therefore, the measure of ∠a is 115°.

In the diagram, TQ and SP intersect at point R.

Thus, $\angle PRQ~\&~\angle TRS$ form vertically opposite angles, and their measures should be equal.

Therefore, $\angle PRQ=\angle TRS=90^o$

In triangle PQR, $\angle PRQ+\angle RQP+\angle QPR=180^o$

$\implies90^o+\angle RQP+50^o=180^o$

$\implies\angle RQP=180^o-140^o=40^o$

Therefore, the measure of ∠b is 40°.

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Answer 2

Answer:

The final answer is

1) a = 105

b) b = 40

Step-by-step explanation:

Given,

∠ATR = 90

∠RPQ = 50

∠AVU = 25

We need to find angles a and b.

So we begin by taking the triangle UAT,

Here we have ∠AVU + ∠VAU + ∠VUA = 180 ( Sum of angles in a triangle)

25 + ∠VAU + ∠VUA = 180

T is a perpendicular line on A, Hence ∠VAU = ∠ATR = 90

25 + 90 + ∠VUA = 180

∠VUA = 75

∠VUA + ∠VUT = 180 ( Exterior angles)

75 + a = 180

a = 105.

Take the quadrant ATRS,

∠TAS = ∠ATR = 90, Then,

∠TRS = ∠ASR = 90 , Since both these angles are adjacent to two 90 angles and adjacent angles in a quadrant is always equal to 180.

In triangle RPQ,

∠RPQ + ∠RQP + ∠ORP = 180 ( Sum of angles in triangle)

Since ∠TSR = 90, Then ∠RPQ = 90 ( Opposites)

∠ORP = 180 - 90 - 50 = 40

b = 40

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