Question

1. which postulate or theorem can be used to prove that △mpn≅△tsr?

a. sas
b. hl
c. sss
d. the triangles cannot be proved to be congruent.

1. name one additional pair of corresponding parts that need to be congruent in order to prove that △car≅△lot by aas.

a. (overline{ac}congoverline{ol})
b. (overline{ar}congoverline{ot})
c. (angle acongangle o)
d. (angle ccongangle l)

1. given △rst≅△hij, (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

1. it is given that △abc≅△fde 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: Recall congruence postulates for 13

SAS (Side - Angle - Side) postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent. From the figure (not shown completely here but assuming the given conditions match SAS criteria), SAS can be used to prove $\triangle MPN\cong\triangle TSR$.

Step2: Analyze AAS for 14

AAS (Angle - Angle - Side) requires two angles and a non - included side to be congruent. For $\triangle CAR\cong\triangle LOT$ by AAS, we need $\angle C\cong\angle L$ as we already have two pairs of angles and need the non - included side condition to be met among the given options.

Step3: Use congruent angle property for 15

Since $\triangle RST\cong\triangle HIJ$, corresponding angles are congruent. So $m\angle S=m\angle I = 80^{\circ}$ and $m\angle R=m\angle H$. In $\triangle RST$, using the angle - sum property of a triangle ($m\angle R+m\angle S+m\angle T=180^{\circ}$), and since $m\angle S = 80^{\circ}$ and $m\angle I=m\angle S = 80^{\circ}$, $m\angle J = 38^{\circ}$, then $m\angle R=180^{\circ}-80^{\circ}-38^{\circ}=62^{\circ}$. Set $4x + 2=62$, then $4x=60$, and $x = 15$.

Step4: Use congruent triangle property for 16

If $\triangle ABC\cong\triangle FDE$, then corresponding sides and angles are congruent. Given $AB = 5$ cm, so $DF=AB = 5$ cm. In $\triangle ABC$, $m\angle C=180^{\circ}-80^{\circ}-40^{\circ}=60^{\circ}$, and since $\triangle ABC\cong\triangle FDE$, $m\angle E=m\angle C = 60^{\circ}$.

Answer:

1. A. SAS
2. D. $\angle C\cong\angle L$
3. A. 15
4. B. $\overline{DF}=5$ cm, $\angle E = 60^{\circ}$