Math Problem Statement

Show that each statement is false by providing a counterexample.

Solution

Let's go through the counterexamples for each statement provided in the image:

(a) If the measures of ∠R, ∠S, and ∠T sum to 180°, then one of the angles must be obtuse.

The given statement is false because three angles can sum to 180° without any of them being obtuse (greater than 90°).

Counterexample:

  • mR=60°m ∠R = 60°
  • mS=60°m ∠S = 60°
  • mT=60°m ∠T = 60°

In this case, all angles are acute (less than 90°), yet they sum to 180°, disproving the claim.


(b) If mWXZ=60°m ∠WXZ = 60°, and point Y is in the interior of ∠WXZ, then mWXY=40°m ∠WXY = 40° and mYXZ=20°m ∠YXZ = 20°.

This statement is false because the interior angles do not necessarily have to match the given values (40° and 20°).

Counterexample:

  • mWXY=30°m ∠WXY = 30°
  • mYXZ=30°m ∠YXZ = 30°

Here, the sum is still 60°, but the division of the angles is different, invalidating the assumption.


(c) If the area of a rectangle is 64, then the length is 8 and the width is 8.

This statement is false because there are other possible dimensions for a rectangle to have an area of 64.

Counterexample:

  • Length = 16
  • Width = 4

Since 16×4=6416 × 4 = 64, this provides a valid counterexample with different dimensions than 8 × 8.


(d) If ∠1 and ∠2 are supplementary angles, then one of them must be acute.

This statement is false because both angles could be right angles (90°), which are not acute.

Counterexample:

  • m1=90°m ∠1 = 90°
  • m2=90°m ∠2 = 90°

Both angles are right angles, and while they are supplementary (sum to 180°), neither is acute.


Would you like further details or clarification on any of the counterexamples?

Here are five related questions to deepen your understanding:

  1. Can a triangle have all angles greater than 90°?
  2. What conditions define supplementary angles?
  3. What are other possible dimensions of a rectangle with an area of 64?
  4. How can you divide an angle into two parts with different measures?
  5. Can an obtuse angle be supplementary to another obtuse angle?

Tip: Always remember, supplementary angles must sum to 180°, but there is no restriction on either being acute or obtuse!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometry
Angles
Supplementary Angles
Area of Rectangles

Formulas

Sum of angles in a triangle: ∠R + ∠S + ∠T = 180°
Area of a rectangle: Area = Length × Width
Supplementary angles: ∠1 + ∠2 = 180°

Theorems

Sum of angles in a triangle theorem
Properties of supplementary angles
Properties of rectangle area

Suitable Grade Level

Grades 7-9