Given: m∠RPQ = 23°; m∠TPR = 85°
If ∠SPR and ∠RPQ are complementary angles, then what is the measure of ∠TPS?
A. 67°
B. 18°
C. 5°
D. 23°
Since ∠SPR and ∠RPQ are complementary angles, we know that:
m∠SPR + m∠RPQ = 90°
Let's call the measure of ∠SPR x. Then we can set up the equation:
x + 23° = 90°
Subtracting 23° from both sides, we get:
x = 67°
Now we need to find the measure of ∠TPS. We know that:
m∠TPR + m∠RPS + m∠SPQ = 180°
Substituting in the values we know, we get:
85° + m∠RPS + 67° = 180°
Simplifying, we get:
m∠RPS = 28°
Now we can find the measure of ∠TPS by using the fact that the angles in a triangle add up to 180°:
m∠TPS = 180° - m∠TPR - m∠RPS
Substituting in the values we know, we get:
m∠TPS = 180° - 85° - 28°
Simplifying, we get:
m∠TPS = 67°
Therefore, the answer is A. 67°.
To find the measure of ∠TPS, we need to determine the measure of ∠SPR, as it is stated that ∠SPR and ∠RPQ are complementary angles.
Given: m∠RPQ = 23°
Since ∠SPR and ∠RPQ are complementary angles, we know that their sum is equal to 90°.
∠SPR + ∠RPQ = 90°
Therefore, we can substitute the given value for ∠RPQ:
∠SPR + 23° = 90°
To find ∠SPR, we isolate it on one side of the equation:
∠SPR = 90° - 23°
∠SPR = 67°
Now that we have determined the measure of ∠SPR, we need to find ∠TPS. We are given that ∠TPR = 85°.
Since ∠TPR and ∠TPS share a common side (TPR), and the sum of the interior angles of a triangle is 180°, we can find ∠TPS by subtracting the sum of ∠RPQ and ∠SPR from 180°:
∠TPS = 180° - (∠RPQ + ∠SPR)
∠TPS = 180° - (23° + 67°)
∠TPS = 180° - 90°
∠TPS = 90°
Therefore, the measure of ∠TPS is 90°.
The correct answer is not provided in the options.