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Two interior angles of a convex quadrilateral ABCD are right angles (a
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08 Jan 2024, 00:03
08 Jan 2024, 00:03
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Two interior angles of a convex quadrilateral ABCD are right angles (all interior angles are less than 180◦ and the two diagonals both lie inside the quadrilateral). The degree measure of ∠ABC is twice the degree measure of ∠BCD.
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
Quantity A |
Quantity B |
Measure of the largest interior angle of quadrilateral ABCD |
120° |
A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
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Two interior angles of a convex quadrilateral ABCD are right angles (a
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Updated on: 10 Jan 2024, 09:26
Updated on: 10 Jan 2024, 09:26
1
The sum of the interior angles of ALL quadrilaterals is 360°.
Given that 2 of the angles are 90° and 2 are not (since one will be twice the measure of the other), it's a trapezoid that looks something like this:
quad0.jpg [ 47.19 KiB | Viewed 589 times ]
where the 2 right angles are adjacent to each other. (The right angles can't be opposite to each other because that would make it a rectangle with all 4 angles being 90°) We also find that the 2 angles must be supplementary (add up to 180°) since 360 - (2*90) = 180.
However, the catch is that we don't know which angles correspond to ∠ABC and ∠BCD. There's 2 possible ways:
(1) 
quad1.jpg [ 33.3 KiB | Viewed 601 times ]
, (2) 
quad2.jpg [ 29.04 KiB | Viewed 594 times ]
With (1), we'd previously found that the 2 non-right angles are supplementary, so 2x + x = 180 --> x = 60°. Since ∠ABC = 2x here, ∠ABC = 120°. The largest angle here is less than Quantity A.
With (2), ∠ABC = 90° = 2x ---> x = 45°. The 2 non-right angles are supplementary, so ∠CDA = 180 - 45 = 135°. The largest angle here is greater than Quantity B.
Since the largest angle could either be less than OR greater than Quantity B, the answer is D.
Given that 2 of the angles are 90° and 2 are not (since one will be twice the measure of the other), it's a trapezoid that looks something like this:
Attachment:
quad0.jpg [ 47.19 KiB | Viewed 589 times ]
where the 2 right angles are adjacent to each other. (The right angles can't be opposite to each other because that would make it a rectangle with all 4 angles being 90°) We also find that the 2 angles must be supplementary (add up to 180°) since 360 - (2*90) = 180.
However, the catch is that we don't know which angles correspond to ∠ABC and ∠BCD. There's 2 possible ways:
(1)
Attachment:
quad1.jpg [ 33.3 KiB | Viewed 601 times ]
Attachment:
quad2.jpg [ 29.04 KiB | Viewed 594 times ]
With (1), we'd previously found that the 2 non-right angles are supplementary, so 2x + x = 180 --> x = 60°. Since ∠ABC = 2x here, ∠ABC = 120°. The largest angle here is less than Quantity A.
With (2), ∠ABC = 90° = 2x ---> x = 45°. The 2 non-right angles are supplementary, so ∠CDA = 180 - 45 = 135°. The largest angle here is greater than Quantity B.
Since the largest angle could either be less than OR greater than Quantity B, the answer is D.
Originally posted by nurirachel on 10 Jan 2024, 09:09.
Last edited by nurirachel on 10 Jan 2024, 09:26, edited 2 times in total.
Last edited by nurirachel on 10 Jan 2024, 09:26, edited 2 times in total.
Re: Two interior angles of a convex quadrilateral ABCD are right angles (a
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10 Jan 2024, 09:18
10 Jan 2024, 09:18
Expert Reply
the images do not show up sir
