Question

1. which postulate or theorem can be used to prove that δmpn≅δtsr? a. sas b. hl c. sss d. the triangles cannot be proved to be congruent. 14. name one additional pair of corresponding parts that need to be congruent in order to prove that δcar≅δlot by aas. a. (overline{ac}congoverline{ol}) b. (overline{ar}congoverline{ot}) c. (angle acongangle o) d. (angle ccongangle l) 15. given δrst≅δhij, (mangle s = 80^{circ}), (mangle j = 38^{circ}), and (mangle r=4x + 2), find the value of x. a. 15 b. 30 c. 120 d. 60 16. it is given that δabc≅δfde and (overline{ab}=5 cm), (angle b = 40^{circ}) and (angle a = 80^{circ}). which of the following is true? a. (overline{df}=5 cm), (angle f = 60^{circ}) b. (overline{df}=5 cm), (angle e = 60^{circ}) c. (overline{de}=5 cm), (angle e = 60^{circ}) d. (overline{df}=5 cm), (angle d = 60^{circ})

Explanation:

Step1: Analyze question 13

Inspect the given triangles for congruence criteria. Since no information about side - angle - side (SAS) congruence is provided in the figure for $\triangle MPN$ and $\triangle TSR$, and we don't have right - angled triangles for HL, and also no information about all three sides being equal for SSS. So the triangles cannot be proved congruent.

Step2: Analyze question 14

For AAS (angle - angle - side) congruence of $\triangle CAR$ and $\triangle LOT$, we already have two angles. We need the non - included side to be congruent. So we need $\overline{AR}\cong\overline{OT}$.

Step3: Analyze question 15

Since $\triangle RST\cong\triangle HIJ$, corresponding angles are equal. So $\angle S=\angle I = 80^{\circ}$, $\angle J=\angle T = 38^{\circ}$, and $\angle R=\angle H$. In a triangle, the sum of angles is $180^{\circ}$. In $\triangle RST$, $\angle R=180^{\circ}-\angle S-\angle T=180 - 80 - 38=62^{\circ}$. Given $\angle R = 4x+2$, then $4x + 2=62$. Subtract 2 from both sides: $4x=60$, divide by 4, $x = 15$.

Step4: Analyze question 16

Since $\triangle ABC\cong\triangle FDE$, corresponding sides and angles are equal. $\overline{AB}=\overline{FD}=5$ cm. $\angle A=\angle F = 80^{\circ}$, $\angle B=\angle D = 40^{\circ}$. Using the angle - sum property of a triangle in $\triangle FDE$, $\angle E=180^{\circ}-\angle F-\angle D=180 - 80 - 40 = 60^{\circ}$.

Answer:

1. D. The triangles cannot be proved to be congruent.
2. B. $\overline{AR}\cong\overline{OT}$
3. A. 15
4. B. $\overline{DF}=5$ cm, $\angle E = 60^{\circ}$