Find pairs of equal triangles and prove their equality.

Explanation:

This task requires identifying pairs of congruent right-angled triangles based on given properties (equal sides, equal angles, right angles) and proving their congruence using established criteria (e.g., Hypotenuse-Leg (HL), Leg-Leg (LL), Hypotenuse-Angle (HA), Leg-Angle (LA)).

Solution:

• Figure 1: Not enough information to determine congruence. It is a rectangle, so triangle ABD is congruent to triangle CDB and triangle ABC is congruent to triangle ADC by LL (sides AB=CD, BC=DA, AC=AC or BD=BD as hypotenuses) and Hypotenuse-Leg (if diagonals are given). Given only right angles, we cannot prove congruence without more information.
• Figure 2: Triangles MKT and NKT are congruent by LL. We have MK = NK (marked with double hash marks), KT is common, and angle MKT = angle NKT = 90 degrees. Therefore, ╮ MKT ≅ ╮ NKT.
• Figure 3: Triangles PK R and SKR are congruent by HA. We have angle P = angle S (marked with arc), angle PKR = angle SKR = 90 degrees, and KR is common. Therefore, ╮ PKR ≅ ╮ SKR.
• Figure 4: Not enough information to determine congruence.
• Figure 5: Triangles PRQ and KRT are congruent by LL. We have PR = RT (marked with double hash marks), RQ = RK (marked with double hash marks), and angle PRQ = angle KRT = 90 degrees. Therefore, ╮ PRQ ≅ ╮ KRT.
• Figure 6: Triangles ACD and BCD are congruent by LA. We have AC = BC (marked with double hash marks), angle ADC = angle BDC = 90 degrees, and CD is common. Therefore, ╮ ACD ≅ ╮ BCD.
• Figure 7: Triangles MRT and NTS are congruent by AAS. We have angle RMT = angle SNT (marked with arcs), angle MTR = angle NTS (vertically opposite angles), and RT = ST (marked with double hash marks). Therefore, ╮ MRT ≅ ╮ NTS.
• Figure 8: Triangles MKN and LNK are congruent by HA. We have MK = LN (marked with double hash marks), angle MKN = angle LNK = 90 degrees, and KN is common. Therefore, ╮ MKN ≅ ╮ LNK.
• Figure 9: Triangles ADC and BFC are congruent by LL. We have AD = BF (marked with double hash marks), DC = FC (marked with double hash marks), and angle ADC = angle BFC = 90 degrees. Therefore, ╮ ADC ≅ ╮ BFC.
• Figure 10: Triangles ADB and CDB are congruent by LL. We have AD = CB (marked with double hash marks), DB is common, and angle ADB = angle CDB = 90 degrees. Therefore, ╮ ADB ≅ ╮ CDB.