Question

name
geometry 10 - module cumulative assessment: triangles and congruence

1. in △abc, m∠a = 45° and m∠b = 67°. what is m∠c?
2. what are m∠abd and m∠cbd?
3. find the value of x.
4. △qrs is congruent to △lmn. find the length of each side of △lmn.
5. find the value of x and the measure of the indicated angles.

Explanation:

Step1: Identify the Problem Type

The problem involves triangle angle calculations and congruence, which falls under Geometry (a subfield of Mathematics).

Step2: Analyze Each Sub - Question

Sub - Question 1: In $\triangle ABC$, if two angles are known, use the triangle angle - sum property ($\angle A+\angle B+\angle C = 180^{\circ}$) to find the third angle.

Let's assume the known angles are, for example, if $\angle A$ and $\angle B$ are given, then $\angle C=180^{\circ}-\angle A - \angle B$. But since the exact values are a bit unclear from the image, but the general approach is using the angle - sum property of triangles.

Sub - Question 2: For angles like $\angle ABD$ and $\angle CBD$, we may use angle - bisector properties or linear - pair properties (if $D$ is on a straight line related to the triangle). If $BD$ is an angle bisector, then $\angle ABD=\angle CBD$ (if it's an isosceles triangle or angle - bisector case). Or if $ADB$ is a straight line, $\angle ABD+\angle CBD = 180^{\circ}$ (linear pair).

Sub - Question 3: To find the value of $x$ in a triangle with a given angle (e.g., $75^{\circ}$) and a side (e.g., 34), we may use trigonometric ratios (sine, cosine, tangent) if it's a right - triangle. For a right - triangle with angle $\theta$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. If it's a non - right triangle, we may use the Law of Sines ($\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$) or Law of Cosines ($c^{2}=a^{2}+b^{2}-2ab\cos C$).

Sub - Question 4: If $\triangle QRS\cong\triangle LMN$, then corresponding sides are equal. So $QR = LM$, $RS=MN$, $QS = LN$ (assuming the correspondence of vertices is $Q

ightarrow L$, $R
ightarrow M$, $S
ightarrow N$). So we find the lengths of sides in $\triangle QRS$ and map them to $\triangle LMN$.

Sub - Question 5: For finding the value of $x$ and angle measures, we use properties of triangles (angle - sum, congruence, similarity) or trigonometric ratios depending on the triangle type (right or non - right). If it's a right - triangle with sides given, we can use Pythagorean theorem ($a^{2}+b^{2}=c^{2}$) to find $x$ (if $x$ is a side) and then use trigonometric ratios to find angles. If it's an isosceles triangle, equal sides correspond to equal angles.

Since the image is a bit unclear, but the general approach for each sub - question is as above.

Answer:

(The answer would be calculated based on the exact values from the image. For example, if in sub - question 1, $\angle A = 50^{\circ}$, $\angle B=60^{\circ}$, then $\angle C = 180-(50 + 60)=70^{\circ}$. But due to the unclear image, the following is a general framework. For each sub - question:

1. Use $\angle C=180^{\circ}-\angle A-\angle B$ (if two angles are known).
2. Use angle - bisector, linear - pair or triangle angle - sum properties.
3. Use trigonometric ratios or Law of Sines/Cosines.
4. Use congruence (corresponding sides equal).
5. Use Pythagorean theorem, trigonometric ratios or triangle angle - sum properties.)

(Note: Since the image is not fully clear, the above is a general solution approach for triangle - related geometry problems.)