Now we see that the triangles have two pairs of congruent sides. Because theses sides included angle are also congruent, we can claim congruence by the SAS (Side-Angle-Side) Congruence Theorem.
What is the Value of x?
To determine x, we will first mark corresponding sides and angles in our triangles. We already know two pairs of corresponding sides, which means we can determine the last pair of corresponding sides as well.
As we can see, θ in the left triangle corresponds to x in the right triangle. Since we know two angles in the left triangle, we can calculate θ and x by using the Triangle Angle Sum Theorem.
θ+28^(∘)+73^(∘)=180^(∘) ⇔ θ = 79^(∘)
Since θ is 79^(∘), we know that x=79^(∘) as well.
b Like in Part A, we will first investigate if the triangles are congruent.
Are the Triangles Congruent?
Let's first determine all the information we can by examining the diagram. From the diagram, we can identify a pair of vertical angles. These are congruent by the Vertical Angles Theorem.
We know that two pairs of angles in the triangles are congruent which means they are similar by the AA (Angle-Angle) Similarity Theorem. However, to claim congruence between the triangles, the angles' included side must be congruent. This is not the case, which means we cannot claim congruence.
c Like in Parts A and B, we will first investigate if the triangles are congruent.
Are the Triangles Congruent?
Examining the diagram, we see that the triangles have two pairs of congruent angles. This means the triangles are similar by the AA Similarity Theorem.
We also see that these are right triangles with congruent hypotenuses. In two congruent right triangles, the hypotenuse of the triangles will always be corresponding sides. Therefore, since a pair of corresponding sides are congruent, the triangles must be congruent.
What is the Value of x?
Let's separate the triangles and identify corresponding sides.
Note that these triangles are 30-60-90 triangles. The ratio of the hypotenuse and legs in such a triangle are as follows.
Since the hypotenuse is 16, we can find the value of a.
2a=16 ⇔ a=8
When we know a we can determine x by equating x+a with asqrt(3), substituting a=8, and solving for x.
Since the triangles are congruent, we know that the unknown leg of the left triangle is 5 units. When we know the length of both legs of the left triangle, we can use the tangent ratio to find the measure of x.
