Question

proof reasons practice
match the statement with the property.

1. if $\triangle abc\cong\triangle xyz$, then $\angle c\cong\angle z$
2. if two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also be congruent.
3. $\overline{fg}\cong\overline{gf}$
4. if $\angle j\cong\angle k$ and $\angle k\cong\angle l$, then $\angle j\cong\angle l$

a. reflexive property
b. transitive property
c. cpctc
d. third angles theorem

find the measure of the indicated angle. then state the angle property you used: alternate interior angles, vertical angles, or linear pair.

1. $m\angle 2 = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$
2. $m\angle 3 = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$
3. $m\angle 6 = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$
4. $m\angle 8 = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$
5. $m\angle a = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$
6. $m\angle f = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$
7. $m\angle e = \underline{\quad\quad}$ reason: $\underline{\quad\quad}$

Explanation:

Step1: Match statements to properties

1. If $\triangle ABC \cong \triangle XYZ$, then $\angle C \cong \angle Z$ corresponds to CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
2. If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also be congruent corresponds to the Third Angles Theorem.
3. $\overline{FG} \cong \overline{GF}$ corresponds to the Reflexive Property (a segment is congruent to itself).
4. If $\angle J \cong \angle K$ and $\angle K \cong \angle L$, then $\angle J \cong \angle L$ corresponds to the Transitive Property.

Step2: Calculate angle 2 measure

$\angle 2$ forms a linear pair with the $54^\circ$ angle. Linear pairs sum to $180^\circ$.
$m\angle 2 = 180^\circ - 54^\circ = 126^\circ$

Step3: Calculate angle 3 measure

$\angle 3$ is a vertical angle to the $54^\circ$ angle, so they are congruent.
$m\angle 3 = 54^\circ$

Step4: Calculate angle 6 measure

$\angle 6$ is a vertical angle to the $140^\circ$ angle, so they are congruent.
$m\angle 6 = 140^\circ$

Step5: Calculate angle 8 measure

$\angle 8$ forms a linear pair with the $41^\circ$ angle. Linear pairs sum to $180^\circ$.
$m\angle 8 = 180^\circ - 41^\circ = 139^\circ$

Step6: Calculate angle a measure

$\angle a$ is a vertical angle to the $83^\circ$ angle, so they are congruent.
$m\angle a = 83^\circ$

Step7: Calculate angle f measure

$\angle f$ forms a linear pair with the $114^\circ$ angle. Linear pairs sum to $180^\circ$.
$m\angle f = 180^\circ - 114^\circ = 66^\circ$

Step8: Calculate angle e measure

$\angle e$ is a vertical angle to the $114^\circ$ angle, so they are congruent.
$m\angle e = 114^\circ$

Answer:

Matching Section:

1. C. CPCTC
2. D. Third Angles Theorem
3. A. Reflexive Property
4. B. Transitive Property

Angle Measure Section:

1. $m\angle 2 = 126^\circ$, Reason: Linear Pair
2. $m\angle 3 = 54^\circ$, Reason: Vertical Angles
3. $m\angle 6 = 140^\circ$, Reason: Vertical Angles
4. $m\angle 8 = 139^\circ$, Reason: Linear Pair
5. $m\angle a = 83^\circ$, Reason: Vertical Angles
6. $m\angle f = 66^\circ$, Reason: Linear Pair
7. $m\angle e = 114^\circ$, Reason: Vertical Angles