We want to determine whether the given figures are , , or neither.
First, let's consider each of these triangles separately.
Diagram a
We will start with triangle from diagram a. We see that the given diagram has a 30^(∘) angle and a 60^(∘) angle. Since the interior angles of a triangle we can subtract the measures of the two angles from 180^(∘) to get the measure of the third angle.
180^(∘) - 30^(∘) - 60^(∘) = 90^(∘)
The missing angle is 90^(∘), so the triangle from diagram a is a . The side opposite the 90^(∘) angle is the and it is 16 units long.
Diagram b
Now, let's consider the triangle from diagram
b. We know all sides in that triangle. Since the previous triangle is a right triangle, let's check if this one is a right triangle. By the , if in a triangle the sum of squares of shorter sides is equal to the square of the longer sides, then the triangle is a right triangle. Our triangle is right if the following equality is true.
10^2 + (10sqrt(3))^2 ? = 20^2
Let's check if we have the equality!
10^2 + (10sqrt(3))^2 ? = 20^2
10^2 + (10 * sqrt(3))^2 ? = 20^2
10^2 + 10^2 * (sqrt(3))^2 ? = 20^2
10^2 + 10^2 *3 ? = 20^2
100+100 * 3? = 400
100+300? = 400
400=400 ✓
θ = cos^(-1) Adjacent/Hypotenuse
θ = cos^(-1) 10/20
θ = cos^(-1) 0.5
θ = 60^(∘)
Diagram c
We will start with triangle from diagram c. We see that the given diagram has a 30^(∘) angle and a 90^(∘) angle. Since the interior angles of a triangle sum to 180^(∘), we can subtract the measures of the two angles from 180^(∘) to get the measure of the third angle.
180^(∘) - 30^(∘) - 90^(∘) = 60^(∘)
The missing angle is 60^(∘), so the triangle from diagram c is a 30^(∘)-60^(∘)-90^(∘) right triangle. In such a triangle, the hypotenuse is twice the length of the leg adjacent to the 60^(∘) angle. In our case, this leg is 8 units, so the hypotenuse is 2* 8 = 16 units.
Diagram d
Now, let's consider the triangle from diagram d. This triangle is a right triangle in which the hypotenuse is twice as long as one of the legs. Let's call the angle between these two sides θ. Next, we will recall the definition of the cosine ratio.
cos θ = Adjacent/Hypotenuse
We can use the inverse cosine function to isolate θ on the left-hand side of the above equation.
θ = cos^(-1) Adjacent/Hypotenuse
We know that the hypotenuse is 10 units and the adjacent leg is 5 units. Let's substitute these values into the above equation and simplify the result.
θ = cos^(-1) Adjacent/Hypotenuse
θ = cos^(-1) 5/10
θ = cos^(-1) 0.5
θ = 60^(∘)
Conclusions
We found that each of the given triangles is a 30^(∘)-60^(∘)-90^(∘) triangle. Therefore, each triangle is similar to any other by the . To find which, if any, are congruent, we can look at the lengths of the hypotenuses. By the two 30^(∘)-60^(∘)-90^(∘) triangle are congruent if they have the same hypotenuse. Otherwise, they are not congruent.
We see that only triangles from diagrams a and c have the same hypotenuse. Therefore, this is the only pair of congruent triangles.
