Question

items 9 - 14. use the figure shown. 9. d is the midpoint of ab. what is m∠a? a 45 c 67.38° b 22.62° d 33.69° 10. what is the length of side a? a 169 b 43 c 13 d 12 11. what is the length of side b? a 169 b 43 c 13 d 12 12. what is m∠bcd? 13. what type of triangle is △abc? select all that apply. a right b equilateral c isosceles d obtuse 14. what type of triangle is △acd? select all that apply. a right b equilateral c isosceles d obtuse items 15 - 18. in the figure, ab is parallel to ce. point f is the midpoint of ae and bc. 15. is m∠baf = m∠cef? explain. a yes; they are vertical angles. b yes; they are alternate interior angles. c yes; they are alternate exterior angles. d no; the triangles are different sizes. 16. which statement is true? a cf = af b cf = ef c ef = bf d af = ef 17. is m∠afb = m∠cfe? explain. a yes; they are vertical angles. b yes; they are alternate interior angles. c yes; they are alternate exterior angles. d no; the triangles are different sizes. 18. is △fba the same size and shape as △fce? explain.

Explanation:

Step1: Recall angle - sum property of a triangle

In $\triangle ABC$, if $D$ is the mid - point of $AB$ and $\angle C = 134.76^{\circ}$, and $\triangle ABC$ is isosceles (since $CD$ is the perpendicular bisector of $AB$). The sum of angles in a triangle is $180^{\circ}$. Let $\angle A=\angle B = x$. Then $x + x+134.76^{\circ}=180^{\circ}$.

Step2: Solve for $\angle A$

$2x=180^{\circ}- 134.76^{\circ}=45.24^{\circ}$, so $x = 22.62^{\circ}$. So for question 9, the measure of $\angle A$ is $22.62^{\circ}$, answer is B.

Step3: Use the Pythagorean theorem in right - triangle $ACD$

In right - triangle $ACD$, $AD = 12$ (since $D$ is the mid - point of $AB = 24$) and $CD = 5$. By the Pythagorean theorem $a=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13$. So for question 10, the length of side $a$ is 13, answer is C.

Step4: Since $\triangle ABC$ is isosceles

$b=a = 13$. So for question 11, the length of side $b$ is 13, answer is C.

Step5: Find $\angle BCD$

$\angle BCD=\frac{1}{2}\angle BCA$. Since $\angle BCA = 134.76^{\circ}$, $\angle BCD = 67.38^{\circ}$. So for question 12, the measure of $\angle BCD$ is $67.38^{\circ}$.

Step6: Classify $\triangle ABC$

$\triangle ABC$ has an angle $\angle C=134.76^{\circ}>90^{\circ}$, so it is an obtuse triangle. Also, since $CD$ is the perpendicular bisector of $AB$, it is isosceles. So for question 13, the answers are C and D.

Step7: Classify $\triangle ACD$

$\triangle ACD$ is a right - triangle. So for question 14, the answer is A.

Step8: Analyze angles in parallel - line situation

Since $AB\parallel CE$ and $F$ is the mid - point of $AE$ and $BC$, $\angle BAF$ and $\angle CEF$ are alternate interior angles. So for question 15, the answer is B.

Step9: Use mid - point property

Since $F$ is the mid - point of $AE$, $AF = EF$. So for question 16, the answer is D.

Step10: Identify vertical angles

$\angle AFB$ and $\angle CFE$ are vertical angles. So for question 17, the answer is A.

Step11: Prove triangle congruence

In $\triangle FBA$ and $\triangle FCE$, $AF = EF$, $\angle AFB=\angle CFE$ (vertical angles) and $BF = CF$ (since $F$ is the mid - point of $BC$). By the Side - Angle - Side (SAS) congruence criterion, $\triangle FBA\cong\triangle FCE$. So for question 18, the answer is yes because of SAS congruence.

Answer:

1. B. $22.62^{\circ}$
2. C. 13
3. C. 13
4. $67.38^{\circ}$
5. C. isosceles, D. obtuse
6. A. right
7. B. Yes; they are alternate interior angles.
8. D. $AF = EF$
9. A. Yes; they are vertical angles.
10. Yes; by SAS congruence.