Home » Test Prep » PSAT Test » Which of the following statements would be sufficient to prove that angle 2 is a right angle, given what is already known?
Refer to the above figure. You are given that AB = 5 and BC = 12. Which of the following statements would be sufficient to prove that angle 2 is a right angle, given what is already known?
I) AC = 13
II) angle BAC and angle BCA are both acute
III) angle 1 is a right angle
A. I and II only
B. None of these
C. II and III only
D. I, II, and III
E. I and III only
Correct Answer: E
Explanation/Reference:
Explanation:
If AC= 13, then ΔABC has short sides AB = 5, BC = 12 and long side AC = 13. Since
52 + 122 = 25 + 144 = 169 = 132, then, by the converse of the Pythagorean Theorem, ΔABC is a right triangle with right angle 2. Statement I is sufficient.
If angle BAC and angle BCA are both acute, we know nothing about angle 2; every triangle has at least two acute angles regardless of type. Statement II tells us nothing.
Angle1 and angle 2 form a linear pair and are therefore supplementary. If one is a right angle, so is the other. Therefore, if angle 1 is a right angle, so is angle 2. Statement III is sufficient.
The correct response is Statement I and III only.
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