Open in App
Log In Start studying!

Select your language

Suggested languages for you:

(a)Let \(\mathrm{ABC}\) be a right triangle with $\mathrm{m} \angle \mathrm{BCA}=90^{\circ}$ and \(\mathrm{m} \angle \mathrm{CAB}=30^{\circ}\). What is $\mathrm{m} \angle \mathrm{ABC} ?$ (b) Prove that in a right triangle the sum of the measures of the angles adjacent to the hypotenuse is \(90^{\circ}\).

Short Answer

Expert verified
\(m \angle ABC = 60^{\circ}\) and the sum of the measures of the angles adjacent to the hypotenuse in a right triangle is always equal to \(90^{\circ}\).
See the step by step solution

Step by step solution

Unlock all solutions

Get unlimited access to millions of textbook solutions with Vaia Premium

Over 22 million students worldwide already upgrade their learning with Vaia!

Step 1: (a) Identifying given angles

In right triangle ABC, we're given that m∠BCA = 90° (making it the right angle) and m∠CAB = 30°. We need to find m∠ABC.

Step 2: (a) Using the Triangle Sum Property

In any triangle, the sum of the interior angles is always equal to 180°. Therefore, m∠ABC + m∠BCA + m∠CAB = 180°.

Step 3: (a) Finding the missing angle

Now we can plug in the values we're given and solve for m∠ABC: m∠ABC + 90° + 30° = 180° m∠ABC + 120° = 180° m∠ABC = 60° So, the measure of angle ABC is 60°.

Step 4: (b) Identifying the hypotenuse

In right triangle ABC, the angle BCA has a measure of 90°, making the side AC the hypotenuse. We now have to prove that the sum of the measures of angles CAB and ABC is always 90°.

Step 5: (b) Using the Triangle Sum Property

Just as in part (a), we will apply the Triangle Sum Property. In right triangle ABC, the sum of the interior angles m∠ABC + m∠BCA + m∠CAB = 180°.

Step 6: (b) Proving the statement

Since we know that m∠BCA = 90°, we can substitute this into the equation: m∠ABC + 90° + m∠CAB = 180° Now, we can rearrange the equation to isolate angles ABC and CAB: (m∠ABC + m∠CAB) + 90° = 180° m∠ABC + m∠CAB = 90° Thus, we have proved that the sum of the measures of the angles adjacent to the hypotenuse in a right triangle is always equal to 90°.

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Access millions of textbook solutions in one place

  • Access over 3 million high quality textbook solutions
  • Access our popular flashcard, quiz, mock-exam and notes features
  • Access our smart AI features to upgrade your learning
Get Vaia Premium now
Access millions of textbook solutions in one place

Join over 22 million students in learning with our Vaia App

The first learning app that truly has everything you need to ace your exams in one place.

  • Flashcards & Quizzes
  • AI Study Assistant
  • Smart Note-Taking
  • Mock-Exams
  • Study Planner
Join over 22 million students in learning with our Vaia App Join over 22 million students in learning with our Vaia App

Recommended explanations on Math Textbooks