Answer:
ASA/AAS
Skills needed: Triangle Geometry
Step-by-step explanation:
1) When looking at Triangles, there are 5 ways to determine Congruence.
---> SSS - When all 3 sides of one triangle are congruent to the other. (The tick marks are used to signify the congruent sides - The sides with 1 tick are congruent to each other, the sides with 2 ticks are congruent to each other, and so on).
---> AAA - When all 3 angles of one triangle are congruent to the other. Again, tick marks are used to signify the congruent angles.
---> SAS - When 2 sides of a triangle are congruent, and the angle in between those two sides of the first triangle is congruent to the second triangle.
---> ASA - When two angles and the side in-between the two angles of one triangle are congruent to the other triangle.
---> AAS - When two angles and one of the two sides not in-between the two angles of the first triangle are congruent to the other triangle.
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As mentioned before, ticks are almost always used to display congruence of sides and angles.
The congruence sign is [tex]\cong[/tex]
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2) In this problem, [tex]\angle XBO \cong \angle RCA[/tex], meaning the angle denoted by XBO (B is the vertex, and X and O are the endpoints) is congruent to the angle denoted by RCA (C is the vertex, R and A are the endpoints)
---> Also, [tex]\angle XOB \cong\angle RAC [/tex] (Angle XOB is congruent to Angle RAC)
2 angles from the first triangle ([tex]\triangle BOX[/tex]) are found to be congruent in the 2nd triangle ([tex]\triangleCAR[/tex][tex]\triangle CAR[/tex])
---------> ALSO:
- [tex]\overline{BO} \cong \overline{CA}[/tex] ---> One side from [tex]\triangle BOX[/tex] is congruent to [tex]\overline{CA}[/tex], which is from the other triangle.
NOTE: This side is also in-between THE TWO ANGLES STATED BEFORE. This means that we have:
ASA ---> Since this has the side in-between the two angles.
ASA is the answer!
[tex]\triangle BOX \cong \triangle CAR \text{ by } \textbf{ASA}[/tex]
