1. The problem asks to determine whether pairs of triangles are congruent by SSS, SAS, ASA, AAS, HL, or none based on given markings of sides and angles. 2. Important rules for triangle congruence: - SSS (Side-Side-Side): All three sides of one triangle are equal to all three sides of another. - SAS (Side-Angle-Side): Two sides and the included angle of one triangle are equal to two sides and the included angle of another. - ASA (Angle-Side-Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another. - AAS (Angle-Angle-Side): Two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another. - HL (Hypotenuse-Leg): In right triangles, if the hypotenuse and one leg are equal, the triangles are congruent. 3. For the first pair of triangles (top-left, pair 1): - Both are right triangles. - Two legs are marked equal. - Since two legs correspond and the triangles are right angled, by HL (Hypotenuse-Leg) or by SSS (if hypotenuse is also equal), the triangles are congruent. 4. For the second pair (top-left, pair 2): - Right triangles with hypotenuse and one leg marked equal. - By HL, these triangles are congruent. 5. For the third pair (top-left, pair 3): - Two triangles with two sides and the included angle equal. - By SAS, these triangles are congruent. 6. For the fourth pair (top-left, pair 4): - Two sides and an adjacent angle equal. - By SAS, these triangles are congruent. 7. For the fifth pair (center-left, pair 5): - Two right triangles flipped horizontally with two legs equal. - By HL, these triangles are congruent. 8. For the sixth pair (center-left, pair 6): - Two triangles with a shared angle and two sets of equal sides. - By SAS, these triangles are congruent. 9. For the seventh pair (center-left, pair 7): - Two triangles tip-to-tip with two pairs of equal sides. - If the included angle is equal, then SAS; otherwise, none. 10. For the eighth pair (center-left, pair 8): - Triangles with three matched sides equal. - By SSS, these triangles are congruent. 11. For the ninth pair (top-right, pair 9): - Right triangles with one side and hypotenuse equal. - By HL, these triangles are congruent. 12. For the tenth pair (top-right, pair 10): - Two equilateral triangles with all sides equal. - By SSS, these triangles are congruent. 13. For the eleventh pair (top-right, pair 11): - Quadrilateral with two triangles inside, four sides and angles equal. - If triangles share two sides and included angle equal, then SAS. 14. For the twelfth pair (top-right, pair 12): - Two right triangles with corresponding hypotenuses and one leg equal. - By HL, these triangles are congruent. Final answer for the first pair (top-left, pair 1): The triangles are congruent by HL. "slug": "triangle congruence", "subject": "geometry", "desmos": {"latex": "", "features": {"intercepts": false, "extrema": false}}, "q_count": 12