1. **Problem statement:** We have an equilateral triangle ABC with sides AB = AC = BC = 6 units. AD is the altitude from vertex A to base BC, and we want to analyze the statements about angles, lengths, and congruence. 2. **Key properties of an equilateral triangle:** - All sides are equal: AB = BC = AC = 6. - All angles are equal: each angle is 60 degrees. - The altitude splits the triangle into two congruent right triangles. 3. **Check each statement:** - Angles B and C are 60 degrees: True, because all angles in an equilateral triangle are 60 degrees. - Calculate length x = AD (altitude): Using Pythagoras theorem in right triangle ABD: $$AB^2 = AD^2 + BD^2$$ Since BD is half of BC (because altitude bisects base in equilateral triangle): $$BD = \frac{6}{2} = 3$$ So, $$6^2 = x^2 + 3^2$$ $$36 = x^2 + 9$$ $$x^2 = 36 - 9 = 27$$ $$x = \sqrt{27} = 3\sqrt{3}$$ - Statement "x = 3\sqrt{3}": True. - Statement "x = 6\sqrt{3}": False. - Triangle ABD congruent to triangle ACD: True, because AD is altitude and bisects BC, so triangles ABD and ACD are mirror images with equal sides and angles. - BD and CD both 3 units long: True, because altitude bisects BC of length 6. 4. **Summary:** - True statements: Angles B and C are 60 degrees, x = 3\sqrt{3}, triangle ABD congruent to triangle ACD, BD and CD are both 3 units long. - False statement: x = 6\sqrt{3}. Final answer: The true statements are: - Angles B and C are 60 degrees. - x = 3\sqrt{3}. - Triangle ABD is congruent to triangle ACD. - BD and CD are both 3 units long.