Question

Write the converse and the contrapositive to the following statements. (a) (Let \(a, b\), and \(c\) be the lengths of sides of a triangle.) If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle. (b) If angle \(A B C\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\)

Short answer

(a) Converse: If right triangle, then \(a^2+b^2=c^2\). Contrapositive: If not right triangle, then \(a^2+b^2\neq c^2\). (b) Converse: If angle \(ABC\) \(>0^{\circ}\) and \(<90^{\circ}\), then acute. Contrapositive: If angle \(ABC\) \(\nless 90^{\circ}\), then not acute.

Step-by-step solution

Understanding the Conditional Statement

Identify the hypothesis ("if" part) and the conclusion ("then" part) of each statement.

Writing the Converse

Form the converse of each statement by switching the hypothesis and conclusion. (a) Original statement: If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle. Converse: If the triangle is a right triangle, then \(a^{2}+b^{2}=c^{2}\).(b) Original statement: If angle \(A B C\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\). Converse: If the measure of angle \(A B C\) is greater than \(0^{\circ}\) and less than \(90^{\circ}\), then \(A B C\) is an acute angle.

Writing the Contrapositive

Form the contrapositive of each statement by negating both the hypothesis and conclusion, then switching them.(a) Original statement: If \(a^{2}+b^{2}=c^{2}\), then the triangle is a right triangle.Contrapositive: If the triangle is not a right triangle, then \(a^{2}+b^{2}eq c^{2}\).(b) Original statement: If angle \(A B C\) is acute, then its measure is greater than \(0^{\circ}\) and less than \(90^{\circ}\).Contrapositive: If the measure of angle \(A B C\) is not greater than \(0^{\circ}\) and less than \(90^{\circ}\), then \(A B C\) is not an acute angle.

Verification

Confirm the correctness of the formulated converses and contrapositives by reviewing the definitions involved in statements.

Key concepts

Converse Statements

In mathematics, a converse statement is derived by reversing the hypothesis and conclusion of a conditional statement. This means that if your original statement is "If P, then Q," the converse would be "If Q, then P." It is important to note that the truth of the original statement does not guarantee the truth of its converse.
For example, consider the statement "If angle ABC is acute, then its measure is greater than 0° and less than 90°." The converse would be "If the measure of angle ABC is greater than 0° and less than 90°, then angle ABC is acute." Both this statement and its converse are true because an acute angle is defined precisely by these boundaries. However, in all cases, you should always verify the truth of the converse separately.

Contrapositive Statements

A contrapositive statement takes a conditional statement and switches both the hypothesis and conclusion while also negating both. It's a bit like creating an inverse of the converse. Given a statement "If P, then Q," its contrapositive would be "If not Q, then not P." Interestingly, a statement and its contrapositive are logically equivalent, meaning if one is true, the other must be true as well.
For instance, starting with the statement "If a triangle has side lengths a, b, and c, and if \( a^2 + b^2 = c^2 \), then the triangle is a right triangle," the contrapositive would be "If a triangle is not a right triangle, then \( a^2 + b^2 eq c^2 \)." Here, the original and its contrapositive both hold true, as per the Pythagorean theorem and its conditions.

Triangle Properties

Triangles have several important properties that are fundamental in geometry. Each triangle has three sides, three angles, and notably, the sum of its internal angles is always 180 degrees. Triangles can be classified based on their angles and sides:

• **Right Triangle**: Has one 90-degree angle, and its sides are connected by the Pythagorean theorem \( a^2 + b^2 = c^2 \).
• **Acute Triangle**: All three angles are less than 90 degrees.
• **Obtuse Triangle**: Has one angle greater than 90 degrees.

These properties help in identifying and solving problems related to triangles. Understanding these classifications is vital when analyzing how side lengths and angles relate in triangle geometry.

Angle Measurement

In geometry, angles are measured in degrees. There are several types of angles based on their measure:

• **Acute Angle**: Measures more than 0 degrees and less than 90 degrees.
• **Right Angle**: Exactly 90 degrees.
• **Obtuse Angle**: Measures more than 90 degrees but less than 180 degrees.
• **Straight Angle**: Exactly 180 degrees.

When working with angle relationships, it's critical to understand how these measurements determine an angle’s classification and affect the geometric shapes formed. Knowing that an acute angle specifically ranges from more than 0° to less than 90° helps in solving problems related to angle position, measurement, and proof. It’s always beneficial to confirm which type of angle you are dealing with to properly apply applicable theorems and properties.