The congruence criteria correspond to the postulates and theorems that state which are the minimum conditions that two or more triangles must meet to be congruent. These are:. If you have two triangles and the three sides of one are respectively congruent with those of the other, then the triangles are congruent. SSS stands for: side, side, side. Two triangles are congruent if two of their sides and the angle between them are respectively equal.
SAS stands for: side, angle, side. If two angles and the side between them are respectively congruent with the same of another triangle, then the triangles are congruent. ASA stands for: angle, side, angle. This theorem states that angles supplement to the same angle are congruent angles, whether they are adjacent angles or not. This theorem states that angles that complement the same angle are congruent angles, whether they are adjacent angles or not.
Let us understand it with the help of the image given below. In this section, we will learn how to construct two congruent angles in geometry.
There are two cases that come up while learning about the construction of congruent angles, and they are:. Step 1- Draw two horizontal lines of any suitable length with the help of a pencil and a ruler or a straightedge. Step 2- Take any arc on your compass, less than the length of the lines drawn in the first step, and keep the compass tip at the endpoint of the line. Draw the arc keeping the lines AB and PQ as the base without changing the width of the compass. Step 3- Keep the compass tip on point D and expand the legs of the compass to draw an arc of any suitable length.
Draw that arc and repeat the same process with the same arc by keeping the compass tip on point S. By now, you have learned about how to construct two congruent angles in geometry with any measurement.
But what if any one angle is given and we have to construct an angle congruent to that? Let's learn it step-wise. Step 2 - Keep compass tip at point B in the given angle and draw an arc by keeping BC as the base and name that point D. Step 3 - With the same width, draw an arc by keeping the compass tip at point Y and name the point at line YZ as O.
Step 4 - Keep compass tip at point D and measure the arc from point D to the point of intersection of the arc at segment AB. Step 5- With the same arc, keep your compass tip at point O and mark a cut at the arc drawn in step 3, and name that point as X.
This is how we can construct an angle congruent to the given angle. This means that the angles that are in the same matching position will have the same angle. Another common test for angle congruence requires a set of parallel lines and a transversal line that slices through the set of parallel lines. For example, lines a and b are parallel, and line l is a transversal that slices through the parallel lines.
When this situation occurs, a handful of congruent angles are formed. Alternate Interior Angles are located in between the two parallel lines, but on alternate sides of the transversal.
Similarly, Alternate Exterior Angles are located on the outside of the parallel lines, and on alternate sides of the transversal. Corresponding Angles are located on the same side of the transversal, and in a similar matching location. Vertical Angles are formed by angles that are opposite of eachother.
Vertical angles, or opposite angles, are commonly used as a proof of congruence. Another category of congruent angles revolves around triangle congruences. Triangle congruence rules are used to prove if two triangles are congruent or not. These rules take into consideration the side lengths and angles of triangles in order to determine congruence. Four criteria are used to determine triangle congruence, and they are conveniently named.
C, then? There are other angle relationships to explore. When you expose these angle relationships, you will establish their truth using a formal proof. For example, you were introduced to the idea of an angle bisector. Well, it turns out that the bisector of an angle divides the angle into two angles, each of which has measure equal to one-half the measure of the original angle. This statement looks a lot like Theorem 9.
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