Question

ricardo draws three right triangles. in each figure, he measures a pair of angles. what is a reasonable conjecture for ricardo to make by recognizing a pattern and using inductive reasoning? in a right triangle, the acute angles are complementary. in a right triangle, the acute angles are congruent. in a right triangle, all of the angles are acute. in a right triangle, all of the angles are right angles.

Explanation:

Step1: Recall triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). For a right - triangle, one angle is \(90^{\circ}\). Let the two acute angles be \(x\) and \(y\). Then \(x + y+90^{\circ}=180^{\circ}\).

Step2: Solve for the sum of acute angles

Subtract \(90^{\circ}\) from both sides of the equation \(x + y+90^{\circ}=180^{\circ}\). We get \(x + y=180^{\circ}-90^{\circ}=90^{\circ}\).

Step3: Analyze the given examples

In the first right - triangle: \(30^{\circ}+60^{\circ}=90^{\circ}\). In the second right - triangle: \(34^{\circ}+56^{\circ}=90^{\circ}\). In the third right - triangle: \(28^{\circ}+62^{\circ}=90^{\circ}\). Two angles are complementary if their sum is \(90^{\circ}\).

The second option “In a right triangle, the acute angles are congruent” is wrong because in the given examples (\(30^{\circ}\) and \(60^{\circ}\), \(34^{\circ}\) and \(56^{\circ}\), \(28^{\circ}\) and \(62^{\circ}\)) the acute angles are not equal. The third option “In a right triangle, all of the angles are acute” is wrong because one angle is \(90^{\circ}\) (a right angle, not acute). The fourth option “In a right triangle, all of the angles are right angles” is wrong because the sum of angles in a triangle would be \(90^{\circ}+90^{\circ}+90^{\circ}=270^{\circ}
eq180^{\circ}\).

Answer:

In a right triangle, the acute angles are complementary.