Is the measure of one of the interior angles of quadrilateral ABCD equal to 60°? Two of the interior angles...

Understanding the Question

We need to determine: Is one of the interior angles of quadrilateral ABCD equal to 60 degrees?

This is a yes/no question. For sufficiency, we need a definitive answer:

• YES: At least one angle equals \(60°\)
• NO: None of the four angles equal \(60°\)

Given Information

• ABCD is a quadrilateral
• The sum of all interior angles in any quadrilateral = \(360°\)

Key Insight

We don't need to find all four angles. If we can confirm even one angle equals \(60°\), we answer YES. If we can rule out \(60°\) for all angles, we answer NO.

Analyzing Statement 1

Statement 1: Two of the interior angles of ABCD are right angles.

What This Tells Us

• Two angles = \(90°\) each (total: \(180°\))
• Remaining two angles must sum to: \(360° - 180° = 180°\)

Testing Different Scenarios

Can the remaining \(180°\) be distributed to give different answers?

Scenario 1: The other two angles are \(60°\) and \(120°\)

• Angles: \(90°, 90°, 60°, 120°\) (sum = \(360°\) ✓)
• Answer: YES (one angle equals \(60°\))

Scenario 2: The other two angles are \(80°\) and \(100°\)

• Angles: \(90°, 90°, 80°, 100°\) (sum = \(360°\) ✓)
• Answer: NO (no angle equals \(60°\))

Conclusion

Since we can construct valid quadrilaterals that give both YES and NO answers, Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

Now analyzing Statement 2 independently (forget Statement 1).

Statement 2: The degree measure of angle ABC is twice the degree measure of angle BCD.

What This Provides

A relationship between two specific angles: \(\mathrm{Angle\,B} = 2 \times \mathrm{Angle\,C}\)

Logical Analysis

This constraint involves only 2 of the 4 angles. Without knowing the other two angles:

Scenario 1: If Angle C = \(30°\)

• Then Angle B = \(60°\)
• Answer: YES (angle B equals \(60°\))

Scenario 2: If Angle C = \(45°\)

• Then Angle B = \(90°\)
• The other two angles are unknown—they could be anything that makes the sum \(360°\)
• We cannot determine if any angle equals \(60°\)

Since different values of C lead to different answers, Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Both Statements

Using both statements together:

• Two angles are \(90°\) (Statement 1)
• \(\mathrm{Angle\,B} = 2 \times \mathrm{Angle\,C}\) (Statement 2)

The Key Question

Which angles are the \(90°\) angles? This matters!

Testing Strategic Scenarios

Scenario 1: The two \(90°\) angles are NOT at B or C

• If angles A and D are both \(90°\)
• Then \(\mathrm{B} + \mathrm{C}\) must equal \(360° - 90° - 90° = 180°\)
• With \(\mathrm{B} = 2\mathrm{C}\): We get \(2\mathrm{C} + \mathrm{C} = 180°\)
• Therefore: \(3\mathrm{C} = 180°\), so \(\mathrm{C} = 60°\)
• Answer: YES (angle C equals \(60°\))

Scenario 2: Angle B is one of the \(90°\) angles

• If \(\mathrm{B} = 90°\), then from \(\mathrm{B} = 2\mathrm{C}\) we get: \(90° = 2\mathrm{C}\)
• Therefore: \(\mathrm{C} = 45°\)
• One other angle is also \(90°\) (from Statement 1)
• The fourth angle = \(360° - 90° - 90° - 45° = 135°\)
• All angles are: \(90°, 90°, 45°, 135°\)
• Answer: NO (no angle equals \(60°\))

Why Combined Statements Are Not Sufficient

We found two valid quadrilateral configurations:

1. Configuration with \(\mathrm{C} = 60°\) → Answer: YES
2. Configuration with all angles being \(90°, 90°, 45°, 135°\) → Answer: NO

Different valid configurations lead to different answers. Therefore, even with both statements combined, we cannot definitively answer whether any angle equals \(60°\).

[STOP - Not Sufficient!]

The Answer: E

The statements together are not sufficient because we can construct valid quadrilaterals that satisfy both constraints yet give different answers to our question.

Answer Choice E: "The statements together are not sufficient."