Answers:
23. Not enough information
25. Congruent (by the HL theorem)
HL = hypotenuse leg
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Explanation for problem 23.
Check out figure 1 shown below.
I have marked the given diagram with light blue and light pink to show which angles are congruent.
Because lines UN and TC are parallel, we know the following alternate interior pairs are congruent.
- angle UNC = angle TCN (light blue)
- angle UNL = angle TCH (light pink)
Therefore, we can establish one link of paired congruent angles of the triangles LNU and TCH. However, we cannot determine any other angle pairings. Why not? Because we don't know if segment UL is parallel to segment TH. If they were parallel, then we could establish something like angle LUN = angle CHT as one alternate interior pairing.
But again, we don't know if UL is parallel to TH. The arrow markers aren't shown for these segments.
Therefore, we simply don't have enough information to determine if the triangles are similar or congruent.
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Explanation for problem 25.
Luckily this problem will provide sufficient info to prove the triangles KSR and ISR are congruent.
We start with the given info that KR = SI due to the identical tickmarks. This represents one node in the flowchart (see figure 2).
Another node is angle RKS = angle RIS because both are right angles, aka 90 degree angles.
A third node is SR = SR because of the reflexive property.
Those three nodes then feed into the final conclusion node that triangle KSR = triangle ISR. The reasoning is "HL theorem" where HL = hypotenuse leg. This theorem works for right triangles only.
Take note how I structured the proof so that it flows from top to bottom. The three nodes side by side are related so they share the same row.
There are many other ways to make the flowchart, so don't feel obligated to use this format. Use whatever works best for you.