Question

question 1(1 point)
if ∠a≅∠d and ∠c≅∠f, which additional statement does not allow you to conclude that △abc≅△def?
question 2(1 point)
what is the value of x?
drawing not to scale
question 3(1 point)
what is the value of x?
drawing not to scale
question 4(1 point)
∠abc≅?

Explanation:

Question 1:

Step1: Recall congruence criteria

We have $\angle A\cong\angle D$ and $\angle C\cong\angle F$. The congruence criteria for triangles are SSS (Side - Side - Side), SAS (Side - Angle - Side), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side) and HL (Hypotenuse - Leg for right - triangles).

Step2: Analyze each option

• Option a: $\overline{BC}\cong\overline{EF}$ gives AAS (two angles and a non - included side).
• Option b: $\overline{AB}\cong\overline{DE}$ gives AAS.
• Option c: $\angle B\cong\angle E$ is redundant as we already have two pairs of angles congruent and this does not help in a new congruence criterion.
• Option d: $\overline{AC}\cong\overline{DF}$ gives ASA.

Question 2:

Step1: Use the angle - sum property of a triangle

The sum of angles in a triangle is $180^{\circ}$. In $\triangle DEF$, we know one angle is $93.5^{\circ}$ and assume it is a non - right triangle. Let the third angle be $y$. Then $x + y+93.5^{\circ}=180^{\circ}$. Also, if we assume the triangle is isosceles or use the fact that the exterior - angle property is not applicable here in a non - relevant way, and since the sum of angles in a triangle is $180^{\circ}$, if we assume the triangle is isosceles with the other non - given angle equal to $x$, then $2x + 93.5^{\circ}=180^{\circ}$.

Step2: Solve for $x$

\[2x=180^{\circ}- 93.5^{\circ}=86.5^{\circ}\]
\[x = 43.25^{\circ}\]

Question 3:

Step1: Recall the isosceles - triangle property

Since two sides of the triangle are equal ($21$), the angles opposite to them are equal. Let the base angles be $x$.

Step2: Use the angle - sum property of a triangle

The sum of angles in a triangle is $180^{\circ}$. So $38^{\circ}+x + x=180^{\circ}$.

Step3: Solve the equation

\[38^{\circ}+2x=180^{\circ}\]
\[2x=180^{\circ}-38^{\circ}=142^{\circ}\]
\[x = 71^{\circ}\]

Question 4:

Step1: Use the congruence of triangles

By SSS (since the markings show equal sides in $\triangle PAM$ and $\triangle BAC$), $\triangle PAM\cong\triangle BAC$. Corresponding angles of congruent triangles are equal. $\angle ABC$ corresponds to $\angle PMN$.

Answer:

Question 1: c. $\angle B\cong\angle E$
Question 2: d. $43.25^{\circ}$
Question 3: a. $71^{\circ}$
Question 4: b. $\angle PMN$