Question

1. which point of concurrency divides each median into a 2:1 ratio? a. centroid b. orthocenter c. circumcenter d. incenter 23. in △xyz and △abc, if ∠x = 40°, ∠y = 70°, and ∠z = 70°, and ∠a = 40°, ∠b = 70°, and ∠c = 70°, which sides correspond to each other? a. xy corresponds to ac, xz corresponds to bc, and yz corresponds to ab. b. correspondences cannot be determined. c. xy corresponds to ab, xz corresponds to ac, and yz corresponds to bc. d. xy corresponds to bc, xz corresponds to ab, and yz corresponds to ac. 24. which of the following is not sufficient to prove two triangles are congruent by sss? a. ab = de, ac = df, bc = ef b. ab = bc, ac = ef, bc = de c. ab = de, bc = df, ac = ef where sides don’t correspond d. ab = de, ac = df, bc = ef where sides correspond correctly

Explanation:

• For question 22: The centroid is the point of concurrency that divides each median of a triangle into a 2:1 ratio.
• For question 23: In similar triangles, corresponding angles are equal and corresponding sides are in proportion. Given the equal - angle measures in \(\triangle XYZ\) and \(\triangle ABC\), the sides opposite equal angles are corresponding. So \(XY\) corresponds to \(AB\), \(XZ\) corresponds to \(AC\), and \(YZ\) corresponds to \(BC\).
• For question 24: The SSS (Side - Side - Side) congruence criterion requires that the three pairs of corresponding sides of two triangles are equal. If the sides don't correspond correctly, it is not sufficient to prove congruence by SSS.

Answer:

1. a. Centroid
2. c. \(XY\) corresponds to \(AB\), \(XZ\) corresponds to \(AC\), and \(YZ\) corresponds to \(BC\).
3. c. \(AB = DE\), \(BC=DF\), \(AC = EF\) where sides don't correspond