![]() ![]() Congruent FiguresĬongruent figures are geometric objects that share the same size and shape in mathematics. Congruence of plane figures is the name of the relationship in use. ![]() If two plane figures, such as line segments, angles, and other figures, are similar in size and shape, they are said to be congruent. The sides of the plane figures are the straight lines or curves that make them up. Some of the plane figures include line segments, curves, or a combination of both line segments and curves. Congruence of Plane figuresĪ geometric figure with no thickness is called a plane figure. The following figures show some of the basic plane shapes: triangles, squares, rectangles, and circles. A vertex is the point where two sides meet, and a side is a straight line that is part of the shape. Different plane shapes have various characteristics, such as various vertices. Plane FiguresĪ plane shape is a closed, 2-D, or flat figure. Congruence is the name given to the relationship between two congruent figures. However, this article will only discuss the congruence of plane figures.įigures that are consistent in size and shape are said to be congruent. The 2-D and 3-D figures are both consistent with each other. Congruent figures in mathematics are those that share the same size and shape. If two shapes are equivalent to one another in all conceivable ways, they are said to be congruent. If two squares have sides of the same length, they are said to be congruent.If the corresponding sides of two rectangles are equal, they are said to be congruent.If the sides and angles of two triangles are the same, they are said to be congruent.Two circles should have the same diameter if they are congruent.You have two right triangles, ABC and RST.Only when two figures have the same size and shape, including their sides, points, angles, etc., can they be said to be congruent. Then it's just a matter of using the SSS Postulate.įigure 12.8 illustrates this situation. If you use the Pythagorean Theorem, you can show that the other legs of the right triangles must also be congruent. Not to mention the fact that a SSA relationship between two triangles is not enough to guarantee that they are congruent. Your plate is so full with initialized theorems that you're out of room. There are several ways to prove this problem, but none of them involve using an SSA Theorem. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the triangles are congruent. Theorem 12.3: The HL Theorem for Right Triangles.Now it's time to make use of the Pythagorean Theorem. You've made use of the perpendicularity of the legs in the last two proofs you wrote on your own. Finally, you know that the two legs of the triangle are perpendicular to each other. You also have the Pythagorean Theorem that you can apply at will. ![]() For example, not only do you know that one of the angles of the triangle is a right angle, but you know that the other two angles must be acute angles. Whenever you are given a right triangle, you have lots of tools to use to pick out important information. Then you'll have two angles and the included side of ABC congruent to two angles and the included side of RST, and you're home free.ĪBC and RST with A ~= R, C ~= T, and ¯BC ~= ¯ST. But wait a minute! Because the measures of the interiorangles of a triangle add up to 180º, and you know two of the angles in are congruent to two of the angles in RST, you can show that the third angle of ABC is congruent to the third angle in RST. If only you knew about two angles and the included side! Then you would be able to use the ASA Postulate to conclude that ABC ~= RST. Given: Two triangles, ABC and RST, with A ~= R, C ~= T, and ¯BC ~= ¯ST.Figure 12.7 Two angles and a nonincluded side of ABC are congruent to two angles and a nonincluded side of RST. ![]()
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