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Chapter 7: Problem 48

PROOF Write a proof. Given ABCD is a parallelogram. \(\angle\)A is a right angle. Prove \(\angle \mathrm{B}, \angle \mathrm{C},\) and \(\angle \mathrm{D}\) are right angles.

Short Answer

Expert verified
The proof confirms that if in a parallelogram one angle is a right angle, then all the other angles are right angles.

Step by step solution

01

Given and Required

In the parallelogram ABCD, it is given that \(\angle\)A is a right angle (90^{\circ}). The objective is to prove that the other angles B, C and D are also right angles.
02

Illustrate the properties of the parallelogram

In any parallelogram, such as our parallelogram ABCD, the opposite angles are equal. This means that \(\angle\)A = \(\angle\)C and \(\angle\)B = \(\angle\)D.
03

Apply the given condition

It's given that \(\angle\)A is a right angle (90^{\circ}). Because \(\angle\)A = \(\angle\)C, we can also conclude that \(\angle\)C is a right angle (90^{\circ}).
04

Use the property of angles in a quadrilateral

In any quadrilateral, the sum of all four angles is 360^{\circ}. Therefore, \(\angle\)A + \(\angle\)B + \(\angle\)C + \(\angle\)D = 360^{\circ}. Substitute the known values: 90^{\circ} + \(\angle\)B + 90^{\circ} + \(\angle\)D = 360^{\circ}. Simplify to: \(\angle\)B + \(\angle\)D = 180^{\circ}.
05

Apply the property of the parallelogram

Since in a parallelogram, \(\angle\)B = \(\angle\)D and \(\angle\)B + \(\angle\)D = 180^{\circ}, this implies that \(\angle\)B = \(\angle\)D = 90^{\circ}. Therefore, \(\angle\)B and \(\angle\)D are also right angles.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallelogram Properties
Understanding the properties of a parallelogram is fundamental in geometry. A parallelogram is a special type of quadrilateral with opposite sides that are parallel. This distinctive feature gives rise to several important properties:
  • Opposite Angles are Equal: This means if one angle is known, the angle directly across will be identical in measure. For instance, in parallelogram ABCD, if \( \angle A = 90^{\circ} \), then \( \angle C = 90^{\circ} \) as well.
  • Consecutive Angles are Supplementary: Any two angles that are adjacent (next to each other) in a parallelogram add up to \( 180^{\circ} \). This is a critical feature used when proving angle measures like in our exercise.
  • Diagonals Bisect Each Other: The diagonals of a parallelogram cut each other in half. However, this property wasn't directly needed in the given proof, it's a beneficial aspect to remember.
These properties are central to recognizing and solving problems involving parallelograms. When given one angle, you can usually predict or deduce the others.
Right Angles
Right angles are a cornerstone concept in geometry, representing a 90-degree angle. They are prevalent across various geometric figures, leading to predictable properties and conclusions:
  • Definition: A right angle is a \( 90^{\circ} \) measure, signifying a perfect square corner.
  • Visual Indicators: Often represented by a small square placed at the vertex, indicating precision in measurement.
  • Impact on Figures: If a quadrilateral, like a rectangle or square, has one right angle, the structure of the shape often forces all corners to share this attribute to maintain parallelism. In our exercise, recognizing \( \angle A \) as a right angle set a chain of reasoning that confirmed all angles were 90 degrees.
Understanding right angles helps in identifying the type of quadrilateral and its properties. Recognizing these angles in figures can lead to expanded proofs and solutions.
Quadrilateral Angle Sum
The sum of the interior angles in any quadrilateral is always \( 360^{\circ} \). This rule is foundational when working with any four-sided figure. Let's delve into this important concept:
  • Reasoning: A quadrilateral can be divided into two triangles. Since each triangle's angles sum up to \( 180^{\circ} \), adding the sums of the two triangles yields \( 360^{\circ} \).
  • Application: In our exercise, knowing one angle was \( 90^{\circ} \) allowed direct application of this principle: \( \angle A + \angle B + \angle C + \angle D = 360^{\circ} \).
  • Solving Problems: Using known angles and supplementary properties, you can solve for unknown angles. In the exercise, knowing three right angles (\( 90^{\circ} \) each) meant that the fourth must also be \( 90^{\circ} \) to fulfill the \( 360^{\circ} \) total.
This sum property simplifies complex quadrilateral problems, as it provides a clear path to determine missing angle measures. Always return to this sum rule when encountered with quadrilateral angle puzzles.

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Most popular questions from this chapter

Can you prove that two parallelograms are congruent by proving that all their corresponding sides are congruent? Explain your reasoning.

Construct any parallelogram and label it \(\mathrm{ABCD}\) Draw diagonals \(\overline{\mathrm{AC}}\) and \(\overline{\mathrm{BD}}\) . Explain how to use paper folding to verify the Parallelogram Diagonals Theorem (Theorem 7.6) for \(\triangle \mathrm{ABCD}\)

REASONINGIn Exercises \(37-40,\) i nd the number of sides for the regular polygon described. Each interior angle has a measure of \(165^{\circ}\)

In Exercises \(37-42,\) the diagonall of rhombus \(\mathrm{ABCD}\) intersect at E. Given that \(\mathrm{m} \angle \mathrm{BAC}\) \(53^{\circ}, \mathrm{DE}\) \(8,\) and \(\mathrm{EC}\) 6 Find the indicated measure. $$ \mathrm{m} \angle \mathrm{DAC} $$

\(\overline{\mathrm{AC}}\) and \(\overline{\mathrm{BD}}\) bisect each other. a. Construct quadrilateral ABCD so that \(\overline{A C}\) and \(\overline{B D}\) are congruent, but not perpendicular. Classify the quadrilateral. Justify your answer. b. Construct quadrilateral \(\mathrm{ABCDso}\) that \(\overline{\mathrm{AC}}\) and \(\overline{\mathrm{BD}}\) are perpendicular, but not congruent. Classify the quadrilateral. Justify your answer.
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