Answer :
Let's solve this step-by-step to determine whether an isosceles right triangle is always a [tex]$45^\circ$[/tex]-[tex]$45^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle.
1. Definition of an Isosceles Triangle:
- An isosceles triangle has two sides that are of equal length.
- The angles opposite these equal sides are also equal.
2. Definition of a Right Triangle:
- A right triangle has one angle that is [tex]$90^\circ$[/tex].
3. Combining Both Definitions:
- An isosceles right triangle has both properties:
- It has one angle of [tex]$90^\circ$[/tex].
- The two other angles must be equal and sum up with the [tex]$90^\circ$[/tex] angle to [tex]$180^\circ$[/tex] (since the sum of all angles in a triangle is [tex]$180^\circ$[/tex]).
4. Calculating the Angles:
- Let the two equal angles each be [tex]$x$[/tex].
- Therefore, the equation is:
[tex]\[ x + x + 90^\circ = 180^\circ \][/tex]
- Simplifying this:
[tex]\[ 2x = 90^\circ \][/tex]
[tex]\[ x = 45^\circ \][/tex]
5. Conclusion:
- Therefore, if a triangle is an isosceles right triangle, the remaining two angles must both be [tex]$45^\circ$[/tex].
- Thus, an isosceles right triangle is always a [tex]$45^\circ$[/tex]-[tex]$45^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle.
Hence, the statement is:
A. True
1. Definition of an Isosceles Triangle:
- An isosceles triangle has two sides that are of equal length.
- The angles opposite these equal sides are also equal.
2. Definition of a Right Triangle:
- A right triangle has one angle that is [tex]$90^\circ$[/tex].
3. Combining Both Definitions:
- An isosceles right triangle has both properties:
- It has one angle of [tex]$90^\circ$[/tex].
- The two other angles must be equal and sum up with the [tex]$90^\circ$[/tex] angle to [tex]$180^\circ$[/tex] (since the sum of all angles in a triangle is [tex]$180^\circ$[/tex]).
4. Calculating the Angles:
- Let the two equal angles each be [tex]$x$[/tex].
- Therefore, the equation is:
[tex]\[ x + x + 90^\circ = 180^\circ \][/tex]
- Simplifying this:
[tex]\[ 2x = 90^\circ \][/tex]
[tex]\[ x = 45^\circ \][/tex]
5. Conclusion:
- Therefore, if a triangle is an isosceles right triangle, the remaining two angles must both be [tex]$45^\circ$[/tex].
- Thus, an isosceles right triangle is always a [tex]$45^\circ$[/tex]-[tex]$45^\circ$[/tex]-[tex]$90^\circ$[/tex] triangle.
Hence, the statement is:
A. True
