Question

Critical Thinking Determine whether each statement is true or false. If true, explain your reasoning. If false, show a counterexample. a. If two triangles are congruent, their perimeters are equal. b. If two triangles have the same perimeter, they are congruent.

Short answer

a. True: Congruent triangles have equal perimeters. b. False: Same perimeter doesn't ensure congruence.

Step-by-step solution

Analyze the Statement a

Statement a claims that if two triangles are congruent, their perimeters are equal. Since congruent triangles are exactly the same in size and shape, all corresponding sides are equal. Therefore, the perimeters, which are sums of their side lengths, must also be equal. Hence, this statement is true.

Analyze the Statement b

Statement b claims that if two triangles have the same perimeter, they are congruent. To verify, consider two triangles with sides (3, 4, 5) and (2, 4, 6). Both have a perimeter of 12 but they are not congruent, as their side lengths are not proportional and they do not have the same shape. This serves as a counterexample, showing the statement is false.

Key concepts

Triangle Perimeter

The perimeter of a triangle is the total length around the triangle. It is calculated by adding up the lengths of all three sides. For instance, if a triangle has sides measuring 3, 4, and 5 units, the perimeter would be 3 + 4 + 5 = 12 units.
Understanding the perimeter is essential because it provides a basis for comparing triangles. However, having the same perimeter doesn't guarantee that two triangles are identical in shape or size.
This difference arises because triangles can have varying internal angles and side length arrangements, leading to distinct triangle shapes even with equal perimeters.

Congruence Criteria

Triangles are congruent when they have identical size and shape. This means all three sides and angles of one triangle are exactly the same as those in the other. The criteria for proving congruence are based on these specific properties.
There are several classic criteria to determine if two triangles are congruent:

• Side-Side-Side (SSS): All three sides of one triangle match the three sides of another.
• Side-Angle-Side (SAS): Two sides and the angle between them in one triangle are equal to those in the other.
• Angle-Side-Angle (ASA): Two angles and the side between them in one triangle are the same as two angles and the corresponding side in the other.
• Angle-Angle-Side (AAS): Two angles and a non-included side are congruent to the corresponding parts of another triangle.

Understanding these criteria helps in determining or denying congruence between triangles.

Counterexamples

Counterexamples are used in mathematics to show that a general statement is false. In this context, a counterexample can be immensely helpful in disproving claims about triangles.
Taking the claim that two triangles with the same perimeter are congruent: a counterexample involves creating two triangles with equal perimeters but different shapes.
For example, consider triangles with side lengths (3, 4, 5) and (2, 4, 6). Both have a perimeter of 12, yet their shapes differ, proving they aren't congruent.
Counterexamples are powerful tools for testing and refining mathematical statements.

Geometric Reasoning

Geometric reasoning involves using logic and mathematical principles to solve problems regarding shapes, their properties, and relationships. It's about understanding why certain geometric statements are true or false.
We apply geometric reasoning to determine triangle congruence or disproving false notions using counterexamples. When examining the congruence between triangles, reasoning through the criteria and scenarios solidifies understanding.
For instance, knowing congruent triangles share identical perimeters uses geometric reasoning, as congruence implies equality of all sides. Conversely, realizing two triangles with equal perimeters may differ in shape involves understanding the variability of triangle configurations.

• This process requires careful analysis of given data and comprehension of triangle properties.
• It encourages critical thinking and helps in deriving accurate conclusions in geometry problems.

By nurturing these reasoning skills, learners can adeptly navigate various geometric challenges.