Math Problem Statement

Prove that triangles or polygons are congruent using theorems and postulates like SSS, SAS, ASA, or AAS. Calculate missing angles or sides.

Solution

The image you uploaded is from a textbook, specifically Section 5.2 on Congruent Polygons. It contains exercises related to triangle congruence and properties of polygons. Below, I'll provide help with some of the problems visible in the image:

Key Exercises:

  1. Problem 9: This involves proving that the quadrilateral ABCDEFGHABCD \cong EFGH using the information provided. To prove congruence between these two quadrilaterals, you need to show that their corresponding sides and angles are congruent. Look for clues like marked angles and side lengths to establish congruence criteria such as SSS (Side-Side-Side) or SAS (Side-Angle-Side).

  2. Problem 10: This asks to prove MNPUST\triangle MNP \cong \triangle UST. Again, you'd use triangle congruence postulates like SSS, SAS, ASA (Angle-Side-Angle), or AAS (Angle-Angle-Side) to demonstrate that these two triangles are congruent.

  3. Problem 11: Involves finding specific missing information. One approach to solving this is through corresponding parts of congruent triangles, which means that once two triangles are proved congruent, all corresponding angles and sides are equal.

  4. Problem 12: Asks to find mZm\angle Z and other missing information. You might use geometric properties or triangle congruence rules to solve this.

  5. Problem 13: Requires finding missing angle measures based on the diagram, particularly angle relationships (e.g., supplementary, complementary, or congruent angles in triangles).

Would you like to go through any specific problem from this image in more detail?

Here are five related questions for deeper practice:

  1. How can you prove two triangles are congruent using the SAS (Side-Angle-Side) postulate?
  2. What is the significance of corresponding parts of congruent triangles (CPCTC)?
  3. How do you calculate the missing angle in a triangle if two angles are given?
  4. What conditions must be met to use the ASA (Angle-Side-Angle) congruence rule?
  5. Can two polygons be congruent if only three sides are given as congruent? Why or why not?

Tip: Always label corresponding parts clearly when proving congruence between polygons. It helps to prevent errors.

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Math Problem Analysis

Mathematical Concepts

Triangle Congruence
Polygon Congruence
Angle Measures

Formulas

Sum of angles in a triangle: 180°
Congruence Postulates: SSS, SAS, ASA, AAS

Theorems

Side-Side-Side (SSS) Congruence Postulate
Side-Angle-Side (SAS) Congruence Postulate
Angle-Side-Angle (ASA) Congruence Postulate
Angle-Angle-Side (AAS) Congruence Postulate

Suitable Grade Level

Grades 8-10