If two angles of a triangle measure 45 degrees each, what is the measure of the third angle?
The Correct Answer and Explanation is :
The measure of the third angle in a triangle where two angles measure 45 degrees each can be found using the fact that the sum of all angles in any triangle is always 180 degrees.
Step-by-Step Explanation:
- Sum of angles in a triangle: The sum of the three interior angles in any triangle is always 180 degrees. This is a fundamental property of triangles.
- Known angles: In this case, two of the angles are given as 45 degrees each. We know that the sum of these two angles is: [
45^\circ + 45^\circ = 90^\circ
] - Finding the third angle: Since the sum of all three angles must be 180 degrees, we can subtract the sum of the two given angles from 180 degrees to find the third angle: [
\text{Third angle} = 180^\circ – 90^\circ = 90^\circ
]
Thus, the third angle of the triangle is 90 degrees.
Explanation:
This triangle is a special type of triangle known as an isosceles right triangle, where the two equal angles are each 45 degrees, and the third angle is 90 degrees. An isosceles triangle has two equal sides, and the angles opposite those sides are also equal. In this case, the two equal angles of 45 degrees are opposite the two equal sides, and the third angle, being a right angle, completes the triangle’s angle sum of 180 degrees.
The right triangle has a specific relationship between its angles and sides, where the hypotenuse is longer than the legs, and the two legs are of equal length in this case.
Now, I will generate an image to visually represent this triangle with angles of 45°, 45°, and 90°.
Here is the visual representation of the isosceles right triangle with two 45° angles and one 90° angle. This illustration clearly shows the equal sides and labeled angles. Let me know if you need any further clarification or details!