Question

two pairs of angles and two pairs of sides?
how could we justify that △abc is congruent to △def?
choose 2 answers:
a side - side - side congruence
b angle - side - angle congruence
c side - angle - side congruence
d hypotenuse - leg congruence

Explanation:

Step1: Analyze Triangle Properties

First, observe that both \(\triangle A'B'C'\) and \(\triangle DEF\) are right triangles (since they have a right angle, as indicated by the right-angle symbol).

Step2: Check Hypotenuse-Leg (HL)

For right triangles, Hypotenuse-Leg (HL) congruence states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent. In \(\triangle A'B'C'\) and \(\triangle DEF\), the hypotenuses: let's check the sides. For \(\triangle A'B'C'\), the hypotenuse (assuming the right angle is at \(A'\) or \(C'\)) – wait, looking at the sides: \(A'B' = 2.5\), \(B'C' = 4.5\)? Wait, no, wait the red triangle \(A'B'C'\): the sides are \(A'B' = 2.5\), \(B'C' = 4.5\)? Wait, no, the blue triangle \(ABC\) has sides \(AB = 5\), \(BC = 9\), right angle at \(A\). The red triangle \(A'B'C'\): right angle, sides \(A'B' = 2.5\), \(B'C' = 4.5\)? Wait, no, the other triangle \(DEF\): \(DE = 2.5\), \(EF = 4.5\), right angle at \(D\). Wait, actually, for right triangles, HL: hypotenuse and leg. Let's see: in \(\triangle A'B'C'\) (right triangle) and \(\triangle DEF\) (right triangle), the legs: \(A'B' = DE = 2.5\), and the hypotenuses? Wait, no, maybe the legs and the included angle? Wait, no, let's check the angles. Wait, the right angle is one angle, then a leg, then the hypotenuse. Wait, also, Side-Angle-Side (SAS): if two sides and the included angle are congruent. The right angle is the included angle between the two legs. So for \(\triangle A'B'C'\) and \(\triangle DEF\), the two legs: one leg is \(2.5\) (e.g., \(A'B'\) and \(DE\)), the other leg? Wait, no, the other side: \(B'C' = 4.5\) and \(EF = 4.5\)? Wait, no, \(DEF\): \(DE = 2.5\), \(DF\) (the leg) – wait, the right angle is at \(D\), so \(DE\) and \(DF\) are legs, \(EF\) is hypotenuse? Wait, no, let's re-express:

For \(\triangle A'B'C'\) (right triangle, right angle at, say, \(A'\)): legs are \(A'B' = 2.5\) and \(A'C'\) (wait, no, the right angle symbol is at \(A'\) or \(C'\)? The diagram shows the right angle at \(A\) for \(ABC\), at \(A'\) for \(A'B'C'\), and at \(D\) for \(DEF\). So \(\triangle A'B'C'\) (right-angled at \(A'\)): legs \(A'B' = 2.5\), \(A'C'\) (unknown), hypotenuse \(B'C' = 4.5\). \(\triangle DEF\) (right-angled at \(D\)): legs \(DE = 2.5\), \(DF\) (unknown), hypotenuse \(EF = 4.5\). So hypotenuse \(B'C' = EF = 4.5\), leg \(A'B' = DE = 2.5\), so HL applies (hypotenuse and leg congruent). Also, SAS: the two legs (2.5 and the other leg? Wait, no, the right angle is the included angle between the two legs. Wait, if we take the two legs: \(A'B' = DE = 2.5\), and the included angle (right angle) is congruent, and the other leg? Wait, no, maybe the legs are \(2.5\) and the other side? Wait, no, maybe I made a mistake. Wait, the correct congruence criteria here: since they are right triangles, HL (Hypotenuse-Leg) is applicable (option D). Also, SAS: because the two legs (the two sides forming the right angle) and the included right angle. So SAS: two sides (legs) and included angle (right angle) congruent, so SAS (option C). Wait, but let's confirm:

- HL: Hypotenuse-Leg: for right triangles, if hypotenuse and one leg are congruent, then triangles are congruent. Here, hypotenuse \(B'C' = EF = 4.5\), leg \(A'B' = DE = 2.5\), so HL (D) is correct.

- SAS: Side-Angle-Side: two sides and the included angle. The two sides are the legs (2.5 and, wait, no, the other leg? Wait, no, the right angle is between the two legs. So if one leg is 2.5, the other leg? Wait, no, maybe the legs are 2.5 and 4.5? Wait, no, the sides: \(A'B' = 2.5\), \(B'C' = 4.5\) (hypotenuse), and \(DE = 2.5\), \(EF = 4.5\) (hypotenuse), and the right angle. Wait, no, SAS: the included angle is between the two sides. So if we have two sides: \(A'B' = DE = 2.5\), the included angle (right angle) is congruent, and the other side: \(B'C' = EF = 4.5\)? No, that's not the included angle. Wait, maybe I messed up. Wait, actually, the right angle is the angle between the two legs. So the two legs are, say, \(A'B'\) (length 2.5) and \(A'C'\) (length, let's calculate: in right triangle, \(A'C' = \sqrt{4.5^2 - 2.5^2}\)? No, wait, no, the other triangle \(DEF\): \(DE = 2.5\), \(DF\) (leg) and \(EF = 4.5\) (hypotenuse). So \(DF = \sqrt{4.5^2 - 2.5^2}\), same as \(A'C'\). So the two legs: \(A'B' = DE = 2.5\), and \(A'C' = DF\) (since both are \(\sqrt{4.5^2 - 2.5^2}\)), and the included angle (right angle) is congruent. So that's SAS: two sides (legs) and included angle (right angle) congruent. So SAS (option C) and HL (option D) are correct. Wait, but let's check the options:

Options:

A. SSS: need three sides, but we only have two sides and a right angle, so SSS is not correct.

B. ASA: angle-side-angle, but we have a right angle, a leg, and hypotenuse, not ASA.

C. SAS: two sides (legs) and included angle (right angle) – yes, because the right angle is between the two legs. So SAS applies.

D. HL: hypotenuse and leg – since they are right triangles, hypotenuse (4.5) and leg (2.5) congruent, so HL applies.

So the correct answers are C and D. Wait, but let's confirm again.

Wait, the problem says "Choose 2 answers". So:

- SAS: two sides and included angle. The two legs (2.5 and the other leg? Wait, no, the two sides are 2.5 (one leg) and 4.5 (hypotenuse)? No, wait, no. Wait, the right angle is between the two legs. So the two legs are, for example, \(A'B'\) (2.5) and \(A'C'\) (let's calculate: in right triangle \(A'B'C'\), right-angled at \(A'\), so \(A'B'^2 + A'C'^2 = B'C'^2\). So \(2.5^2 + A'C'^2 = 4.5^2\). \(A'C'^2 = 20.25 - 6.25 = 14\), so \(A'C' = \sqrt{14}\). Similarly, in \(\triangle DEF\), right-angled at \(D\), \(DE = 2.5\), \(EF = 4.5\), so \(DF^2 = 4.5^2 - 2.5^2 = 14\), so \(DF = \sqrt{14}\). So the two legs are \(A'B' = DE = 2.5\) and \(A'C' = DF = \sqrt{14}\), and the included angle (right angle) is congruent. So that's SAS: two sides (legs) and included angle (right angle) congruent. So SAS (C) is correct.

HL: hypotenuse \(B'C' = EF = 4.5\) and leg \(A'B' = DE = 2.5\), so HL (D) is correct.

So the two correct answers are C (SAS) and D (HL). Wait, but let's check the options again. The options are:

A. SSS – no.

B. ASA – no.

C. SAS – yes.

D. HL – yes.

So the correct answers are C and D.

Answer:

C. Side-angle-side congruence, D. Hypotenuse-leg congruence