Question:
If two interior angles of a quadrilateral ABCD are right angles and the degree measure of \(\angle\)ABC is twice the degree measure of \(\angle\)BCD, what could be the measure of the largest interior angle of quadrilateral ABCD?
Indicate all such answers.
[Note: Select one or more answer choices]
If two interior angles of a quadrilateral ABCD are right angles and the degree measure of \(\angle\)ABC is twice the degree measure of \(\angle\)BCD, what could be the measure of the largest interior angle of quadrilateral ABCD?
Indicate all such answers.
[Note: Select one or more answer choices]
Indicate all such answers.
[Note: Select one or more answer choices]
Show Hint
In geometry problems with unspecified labels (e.g., "two angles are right angles"), it is crucial to systematically consider all distinct cases. The main cases for a quadrilateral are that the right angles are adjacent or opposite. Be sure to check how the given relationships interact with each case.
Updated On: Oct 3, 2025
- 90°
- 105°
- 120°
- 135°
- 150°
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
The sum of the interior angles in any quadrilateral is 360°. We are given information about the angles and need to find the largest possible angle. The problem is ambiguous about which two angles are the right angles, so we must consider all distinct cases. We will assume the quadrilateral is convex, meaning all interior angles are less than 180°.
Step 2: Key Formula or Approach:
1. The sum of angles is \(\angle A + \angle B + \angle C + \angle D = 360^\circ\).
2. We are given the relation \(\angle B = 2 \times \angle C\).
3. Two of the four angles are 90°. We need to check all possible arrangements for the two right angles.
- Case 1: The right angles are not \(\angle B\) or \(\angle C\).
- Case 2: One of the right angles is \(\angle B\) or \(\angle C\).
- Case 3: Both right angles are \(\angle B\) and \(\angle C\).
4. Solve for the angles in each valid case and find the largest angle.
Step 3: Detailed Explanation:
Let the angles be A, B, C, and D. We are given B = 2C. The sum A + B + C + D = 360°.
Case I: The right angles are A and D.
So, A = 90° and D = 90°.
The sum becomes \(90 + B + C + 90 = 360\), which simplifies to \(B + C = 180\).
Substitute \(B = 2C\) into this equation:
\(2C + C = 180 \implies 3C = 180 \implies C = 60^\circ\)
Then \(B = 2C = 2 \times 60 = 120^\circ\).
The angles are {90°, 120°, 60°, 90°}. The largest angle is 120°.
Case II: The right angles are adjacent, one of them involved in the B = 2C relation.
Let's assume the right angles are A and B. So, A = 90° and B = 90°.
The relation \(B = 2C\) becomes \(90 = 2C\), which gives \(C = 45^\circ\).
Now find the fourth angle D from the sum:
\(90 + 90 + 45 + D = 360 \implies 225 + D = 360 \implies D = 135^\circ\).
The angles are {90°, 90°, 45°, 135°}. The largest angle is 135°.
(If we assume the right angles are B and D, the result is the same: B = 90, D = 90 → C = 45, A = 135. Largest angle is 135°).
Case III: The right angles are adjacent, both involved in the B = 2C relation.
This means B = 90° and C = 90°. The relation \(B = 2 \times 90\) becomes \(90 = 180\), a contradiction. This case is impossible.
Case IV: The right angles involve C and an angle not B.
Let's assume the right angles are C and D. So, C = 90° and D = 90°.
The relation \(B = 2C\) becomes \(B = 2 \times 90 = 180^\circ\). An interior angle of 180° implies a degenerate quadrilateral, which is typically excluded unless specified. If allowed, 180° would be the largest angle. Assuming convex quadrilaterals, we discard this case.
(The same happens if the right angles are A and C, leading to B = 180°).
Based on the valid, non-degenerate cases, the possible values for the largest angle are 120° and 135°.
Step 4: Final Answer:
The possible measures for the largest interior angle are 120° and 135°. These correspond to options (C) and (D).
The sum of the interior angles in any quadrilateral is 360°. We are given information about the angles and need to find the largest possible angle. The problem is ambiguous about which two angles are the right angles, so we must consider all distinct cases. We will assume the quadrilateral is convex, meaning all interior angles are less than 180°.
Step 2: Key Formula or Approach:
1. The sum of angles is \(\angle A + \angle B + \angle C + \angle D = 360^\circ\).
2. We are given the relation \(\angle B = 2 \times \angle C\).
3. Two of the four angles are 90°. We need to check all possible arrangements for the two right angles.
- Case 1: The right angles are not \(\angle B\) or \(\angle C\).
- Case 2: One of the right angles is \(\angle B\) or \(\angle C\).
- Case 3: Both right angles are \(\angle B\) and \(\angle C\).
4. Solve for the angles in each valid case and find the largest angle.
Step 3: Detailed Explanation:
Let the angles be A, B, C, and D. We are given B = 2C. The sum A + B + C + D = 360°.
Case I: The right angles are A and D.
So, A = 90° and D = 90°.
The sum becomes \(90 + B + C + 90 = 360\), which simplifies to \(B + C = 180\).
Substitute \(B = 2C\) into this equation:
\(2C + C = 180 \implies 3C = 180 \implies C = 60^\circ\)
Then \(B = 2C = 2 \times 60 = 120^\circ\).
The angles are {90°, 120°, 60°, 90°}. The largest angle is 120°.
Case II: The right angles are adjacent, one of them involved in the B = 2C relation.
Let's assume the right angles are A and B. So, A = 90° and B = 90°.
The relation \(B = 2C\) becomes \(90 = 2C\), which gives \(C = 45^\circ\).
Now find the fourth angle D from the sum:
\(90 + 90 + 45 + D = 360 \implies 225 + D = 360 \implies D = 135^\circ\).
The angles are {90°, 90°, 45°, 135°}. The largest angle is 135°.
(If we assume the right angles are B and D, the result is the same: B = 90, D = 90 → C = 45, A = 135. Largest angle is 135°).
Case III: The right angles are adjacent, both involved in the B = 2C relation.
This means B = 90° and C = 90°. The relation \(B = 2 \times 90\) becomes \(90 = 180\), a contradiction. This case is impossible.
Case IV: The right angles involve C and an angle not B.
Let's assume the right angles are C and D. So, C = 90° and D = 90°.
The relation \(B = 2C\) becomes \(B = 2 \times 90 = 180^\circ\). An interior angle of 180° implies a degenerate quadrilateral, which is typically excluded unless specified. If allowed, 180° would be the largest angle. Assuming convex quadrilaterals, we discard this case.
(The same happens if the right angles are A and C, leading to B = 180°).
Based on the valid, non-degenerate cases, the possible values for the largest angle are 120° and 135°.
Step 4: Final Answer:
The possible measures for the largest interior angle are 120° and 135°. These correspond to options (C) and (D).
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