[tex]\large\textsf{PART 1. Find x.}[/tex]
[tex]\textsf{Number 1:}[/tex]
- x is a base angle of an isosceles triangle.
- The other base angle is 73°.
- Using the Isosceles Triangle Theorem,
[tex]\qquad \qquad\qquad\red{\boxed{\tt x=73°}}[/tex]
[tex]\\[/tex]
[tex]\textsf{Number 2:}[/tex]
- 65° is a base angle of an isosceles triangle.
- ∠2 is a vertex angle.
- Using the Isosceles Triangle Theorem,
[tex]\qquad\tt 65+65+m\angle 2=180 \\ \\ \qquad\tt 65+65+8x - 6 = 180 \\ \\ \qquad\tt 124+8x=180 \\ \\ \qquad\tt 8x=180-124 \\ \\ \qquad\tt 8x=56 \\ \\ \qquad\tt \frac{8x}{8} = \frac{56}{8} \\ \\ \qquad\red{\boxed{\tt x=7}}[/tex]
[tex]\\[/tex]
[tex]\large\textsf{PART 2. Fill in the blank.}[/tex]
*The two-column proof is attached above.
[tex]\\[/tex]
[tex]\large\textsf{PART 3.}[/tex]
[tex]\textsf{For A:}[/tex]
Answer: [tex]\red{\tt \overline{LJ}\cong \overline{LM}, \overline{LK}\cong \overline{LK}}[/tex] or
[tex]\qquad\qquad[/tex][tex]\red{\tt \overline{LJ}\cong \overline{LM}, \overline{JK}\cong \overline{MK}}[/tex]
Explanation:
☞ HL Congruence Theorem requires that the hypotenuse and a leg of the first right triangle is congruent to the corresponding hypotenuse and a leg of the second right triangle.
☞ Just as long as the hypotenuse of the two triangles are congruent, any of the two corresponding legs can suffice the HL Congruence Theorem.
[tex]\\[/tex]
[tex]\textsf{For B:}[/tex]
Answer: [tex]\red{\tt LL\: Congruence\:Theorem}[/tex]
Explanation:
☞The LL Congruence Theorem requires that two corresponding legs of the two right triangles are congruent.
- First corresponding legs: Given that K is the midpoint of JM, we can prove that JK≅MK by Definition of Midpoint.
- Second corresponding legs: LK≅LK by Reflexive Property of Equality.
[tex]\\[/tex]
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