1-28 Conjectures Geometry Zahn

The Corresponding Angles Conjecture states that if two parallel lines are cut by a transversal, then corresponding angles are congruent. This means that angles in the same position on each side of the transversal will have equal measures.

The correct relationship described by the C-3b Alternate Interior Angles Conjecture (AIA) is that alternate interior angles are congruent when two parallel lines are cut by a transversal.

The correct conjecture states that alternate exterior angles are congruent when two parallel lines are cut by a transversal.

The C-3 Parallel Lines Conjecture states that when two parallel lines are cut by a transversal, specific angle relationships are formed between corresponding angles, alternate interior angles, and alternate exterior angles.

The Converse of the Parallel Lines Conjecture states that if certain pairs of angles formed by the transversal are congruent, then the lines are parallel. Therefore, the incorrect answers provided do not align with the conditions required for the lines to be parallel according to this conjecture.

The C-5 Perpendicular Bisector Conjecture states that a point lies on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints, leading to the explanation of the correct answer.

The Converse of the Perpendicular Bisector Conjecture states that if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. This is an important geometric principle that helps understand the relationship between points, segments, and perpendicular bisectors.

The correct answer states that the shortest distance is along the perpendicular segment, not the diagonal, horizontal, or vertical segment.

The correct answer states that if a point is on the bisector of an angle, then it is equidistant from the sides of the angle. This is a fundamental property of angle bisectors in geometry.

The C-10 Perpendicular Bisector Concurrency Conjecture specifically applies to triangles and not other geometric figures such as quadrilaterals, circles, or line segments.

The C-11 Altitude Concurrency Conjecture specifically refers to the concurrence of the altitudes in a triangle, not to other properties such as equal sides, angle sums, or medians.

The C-12 Circumcenter Conjecture states that the circumcenter of a triangle is the point that is equidistant from each of the three vertices of the triangle.

The C-13 Incenter Conjecture states that the incenter of a triangle is equidistant from the sides. This means that the incenter is the same distance away from each side of the triangle.

The C-14 Median Concurrency Conjecture states that the three medians of a triangle intersect at a single point, known as the centroid of the triangle.

The correct answer describes the specific property of a C-15 centroid, where it relates to the division of medians in a triangle.

The correct answer is based on the Angle-Angle-Angle (AAA) congruence theorem in geometry, which states that if two angles of one triangle are equal in measure to two angles of another triangle, then the third angle in each triangle must also be equal.

The Isosceles Triangle Conjecture specifically states that if a triangle is isosceles, then its base angles are congruent. This is a fundamental property of isosceles triangles.

The correct answer is the statement that directly relates to the Isosceles Triangle Conjecture and its converse.

The C-21 Triangle Inequality Conjecture states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side, highlighting the relationship between the sides of a triangle for it to be a valid geometric shape.

The correct answer is that in a triangle, if one side is longer than another side the angle opposite the longer side is larger than the angle opposite the shorter side, not smaller. There is a clear relationship between the length of the sides and the corresponding angles, which is known as the C-22 Side-Angle Inequality Conjecture.

The C-23 Triangle Exterior Angle Conjecture states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. This means that when you extend one side of a triangle, the exterior angle formed is equal to the sum of the two interior angles that are not adjacent to it.

The SSS Congruence Conjecture specifically deals with the side-side-side criterion for triangle congruence.

The correct answer explains the ASA congruence criterion for triangles, where the angles and included side must be congruent for the triangles to be congruent. The incorrect answers provide common misconceptions or incorrect statements related to triangle congruence.

The C-27 SAA Congruence Conjecture specifically refers to the situation where two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle in order to conclude that the two triangles are congruent. The other three options do not accurately represent this concept.