Answer:
- ∠PQR measures 40°
- x = 7
- They form concentric circles
- Yes, according to the triangle inequality
Step-by-step explanation:
1. ∠PQR is an alternate interior angle with the one marked 40°, so it has the measure 40°.
∠PRQ is an alternate interior angle with the one marked 60°, so it has the measure 60°.
Among the answer choices, the one describing ∠PQR as 40° is the only correct one.
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2. ∠BAD = 2×∠BAE
130 = 2(9x + 2) . . . substitute the given expressions
65 = 9x + 2 . . . . . divide by 2
63 = 9x . . . . . . . . .subtract 2
7 = x . . . . . . . . . . . .divide by 9
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3. Think again. Anything being rotated follows a <em>circular path</em>. Circular paths with different radii and the same center are concentric circles.
Straight lines connecting the pre-image and image points will be parallel (and different lengths), but the question is concerned with <em>paths</em>, not <em>endpoints</em>.
Why the path is described as "concentric circles," we're not sure. The path for a 90° rotation will be a 90° arc. It is perfectly reasonable to describe the paths of the two points as <em>concentric arcs</em>, rather than <em>concentric circles</em>. See the attachment.
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4. Correct. The triangle inequality requires the sum of the two shortest side lengths exceed the longest side length. Here, that means 2+5 > 5 (true). The "toothpicks" meet the requirements of the triangle inequality, so will make a triangle.
The "triangle sum theorem" has to do with angles, not side lengths.
