Unit 7 Polygons And Quadrilaterals Homework 2 Parallelograms Answer Key

Unit 7 Polygons and Quadrilaterals Homework 2: Parallelograms – Answer Key & Comprehensive Guide

This comprehensive guide delves into the concepts of parallelograms, providing you with an answer key for Homework 2 of Unit 7 (Polygons and Quadrilaterals) and a detailed explanation of the key properties and theorems related to parallelograms. We’ll cover various problem-solving techniques and provide additional practice problems to solidify your understanding. This guide is designed to help you master the topic and improve your geometry skills.

Understanding Parallelograms: A Foundation

Before diving into the answer key, let's establish a strong foundation in parallelogram properties. A parallelogram is a quadrilateral (a four-sided polygon) with two pairs of parallel sides. This seemingly simple definition leads to several crucial characteristics:

Key Properties of Parallelograms:

• Opposite sides are parallel: This is the defining characteristic. Sides AB and CD are parallel, as are sides BC and AD.
• Opposite sides are congruent: The lengths of opposite sides are equal. AB = CD and BC = AD.
• Opposite angles are congruent: The measures of opposite angles are equal. ∠A = ∠C and ∠B = ∠D.
• Consecutive angles are supplementary: Angles that share a side add up to 180°. ∠A + ∠B = 180°, ∠B + ∠C = 180°, ∠C + ∠D = 180°, and ∠D + ∠A = 180°.
• Diagonals bisect each other: The diagonals of a parallelogram intersect at a point that divides each diagonal into two equal segments.

Homework 2: Parallelograms – Answer Key & Explained Solutions

Note: Since I do not have access to your specific homework assignment, I cannot provide the exact answer key. However, I will illustrate how to solve various types of problems related to parallelograms, providing examples that mirror common homework questions.

Example Problem 1: Finding Missing Angles

Problem: In parallelogram ABCD, ∠A = 110°. Find the measure of ∠B, ∠C, and ∠D.

Solution:

• ∠B: Consecutive angles are supplementary, so ∠A + ∠B = 180°. Therefore, ∠B = 180° - 110° = 70°.
• ∠C: Opposite angles are congruent, so ∠C = ∠A = 110°.
• ∠D: Opposite angles are congruent, so ∠D = ∠B = 70°.

Example Problem 2: Finding Missing Side Lengths

Problem: In parallelogram EFGH, EF = 8 cm and FG = 12 cm. Find the lengths of GH and EH.

Solution:

• GH: Opposite sides are congruent, so GH = EF = 8 cm.
• EH: Opposite sides are congruent, so EH = FG = 12 cm.

Example Problem 3: Using the Diagonals

Problem: The diagonals of parallelogram ABCD intersect at point E. If AE = 5x + 2 and EC = 7x - 4, find the value of x and the length of AC.

Solution:

• Diagonals bisect each other: AE = EC. Set up the equation: 5x + 2 = 7x - 4.
• Solve for x: Subtract 5x from both sides: 2 = 2x - 4. Add 4 to both sides: 6 = 2x. Divide by 2: x = 3.
• Length of AC: AC = AE + EC = (5x + 2) + (7x - 4) = 12x - 2. Substitute x = 3: AC = 12(3) - 2 = 34.

Example Problem 4: Proving Parallelograms

Problem: Given quadrilateral RSTU, prove that it is a parallelogram if RS || TU and RS ≅ TU.

Solution:

This problem requires using theorems about parallelograms. If you have a quadrilateral where one pair of opposite sides is both parallel and congruent, then it’s a parallelogram. This is a theorem you'll likely have learned in your unit.

Example Problem 5: Coordinate Geometry and Parallelograms

Problem: Points A(1,2), B(4,3), C(6,0), and D(3,-1) are vertices of a quadrilateral. Determine if ABCD is a parallelogram.

Solution:

This involves using the distance formula and slope formula. You need to check if opposite sides have equal lengths (distance formula) and equal slopes (meaning they are parallel). If both conditions are met, it's a parallelogram.

Advanced Concepts and Problem-Solving Strategies

Moving beyond the basics, let's explore some more advanced concepts related to parallelograms:

Special Types of Parallelograms:

• Rectangles: Parallelograms with four right angles.
• Rhombuses: Parallelograms with four congruent sides.
• Squares: Parallelograms that are both rectangles and rhombuses (four congruent sides and four right angles).

Understanding the relationships between these special parallelograms and their properties will help you solve more complex problems. For example, knowing that a square is a special type of rectangle and rhombus helps deduce its properties.

Problem-Solving Tips:

• Draw diagrams: Always start by creating a clear diagram to visualize the problem. Label all known information.
• Identify key properties: Recall the properties of parallelograms relevant to the problem.
• Use algebraic techniques: Set up equations to solve for unknown variables.
• Break down complex problems: If a problem seems overwhelming, break it down into smaller, more manageable parts.
• Check your work: Once you've found a solution, review your steps and ensure your answer makes sense in the context of the problem.

Practice Problems

To reinforce your understanding, here are some additional practice problems:

1. In parallelogram KLMN, ∠K = (3x + 10)° and ∠M = (5x - 20)°. Find the value of x and the measure of each angle.

2. The diagonals of parallelogram PQRS intersect at point T. If PT = 2y + 3 and TQ = 5y - 9, find the value of y and the length of PR.

3. Prove that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

4. Points E(2,1), F(5,4), G(7,2), and H(4,-1) are vertices of a quadrilateral. Determine if EFGH is a parallelogram using coordinate geometry.

By working through these practice problems and thoroughly understanding the concepts and properties discussed in this guide, you'll be well-prepared to tackle any parallelogram problems you encounter, building a solid foundation in geometry. Remember to always approach problems systematically, breaking them down into smaller steps and referencing the key properties of parallelograms to guide your solutions. Good luck!