The figure shows circle C with inscribed quadrilateral JKLM.
Which statements are sufficient to know that all four angles of JKLM must be right angles?
A. <j=<L
B. <J and <K are right angles
c. m2J + m2L = 180° and m/K + m/M = 180°
D. JKL and LMJ are semicircles
E. JKL and KLM are semicircles
F. IMJ is a semicircle and m2J + m2K = 180°
A, B, C
Based on the given options, the statements that are sufficient to know that all four angles of JKLM must be right angles are:
B. <J and <K are right angles
and
D. JKL and LMJ are semicircles
If <J and <K are right angles (statement B), then angles JKL and KLM will also be right angles since they are adjacent angles to <J and <K. This covers two angles of JKLM being right angles.
If JKL and LMJ are semicircles (statement D), it implies that they are both 180° angles. Since JKLM is inscribed in circle C, the sum of the opposite angles in an inscribed quadrilateral is 180°. Therefore, if JKL and LMJ are both 180° angles, the other two angles of JKLM must also be right angles.
So, statement B combined with statement D is sufficient to know that all four angles of JKLM must be right angles.
To determine which statements are sufficient to know that all four angles of quadrilateral JKLM must be right angles, we can use our knowledge of the properties of inscribed quadrilaterals and circles.
Statement A: <J = <L
This statement alone is not sufficient to conclude that all four angles of JKLM must be right angles. It only implies that <J and <L are congruent angles, but it does not provide information about the other two angles.
Statement B: <J and <K are right angles
This statement alone is sufficient to conclude that all four angles of JKLM must be right angles. If <J and <K are both right angles, it means that the opposite angles of JKLM are also right angles, making all four angles right angles.
Statement C: m2J + m2L = 180° and m/K + m/M = 180°
This statement alone is not sufficient to conclude that all four angles of JKLM must be right angles. The given conditions only state that the sum of the measures of angles <J and <L is 180°, and the sum of the measures of angles <K and <M is also 180°. However, this does not necessarily mean that all four angles are right angles. They could be any pair of angles whose measures add up to 180°.
Statement D: JKL and LMJ are semicircles
This statement alone is sufficient to conclude that all four angles of JKLM must be right angles. If both JKL and LMJ are semicircles, then the sum of all the angles at the vertices of the quadrilateral would be 360° (since the angles of a semicircle add up to 180°). In a 360° figure, all four angles must be right angles.
Statement E: JKL and KLM are semicircles
This statement alone is not sufficient to conclude that all four angles of JKLM must be right angles. Although JKL and KLM being semicircles implies that three of the angles in JKLM are right angles, it doesn't guarantee that the fourth angle will also be a right angle. There could be a different arrangement where the fourth angle is not a right angle.
Statement F: IMJ is a semicircle and m2J + m2K = 180°
This statement alone is not sufficient to conclude that all four angles of JKLM must be right angles. While IMJ being a semicircle implies that angle <J is a right angle, the given condition about the measures of angles <J and <K adds up to 180° doesn't provide information about the other two angles.
Therefore, the statements that are sufficient to know that all four angles of JKLM must be right angles are:
- Statement B: <J and <K are right angles
- Statement D: JKL and LMJ are semicircles.