In this article, we will be looking at some of the basic theorems of straight lines and triangles. In most of the Geometry based questions in competitive tests, theorems offer the starting point of the solution. Over the next few articles, we will be exploring all the theorems regarding lines, triangles, and circles that one needs to know from CAT preparation point of view.
Theorem 1: If a straight line stands on another straight line then the sum of the two adjacent angles is 180°.
This is a basic theorem and can be illustrated with the help of this diagram. In this case, AB is a straight line and PQ stands on it. Hence, ∠ APQ + ∠ QPB = 180° The converse of this theorem is also true. Hence, If the sum of two adjacent angles AOC and COB with the common arm OC is two right angles, then OA and OB are supplementary rays.
Theorem 2: If two straight lines intersect, the vertically opposite angles so formed are equal.
In the diagram above, lines AB and PQ intersect each other at point O. Hence,
∠ AOQ = ∠ POB and ∠ AOP = ∠ QOB
The proof of this theorem is quite simple. If ∠ AOQ = x and ∠ AOP = y, x + y = 180° (refer the first theorem). As ∠ AOP + ∠ POB = 180°, we will get ∠ POB = x. Hence, ∠ AOQ = ∠ POB
Theorem 3: The sum of any two angles of a triangle is less than 180°.
As the third angle of a triangle can’t be zero, the sum of the remaining two angles will always be less than 180°. This theorem also gives two corollaries.
1. Any exterior angle of a triangle is greater than any of the two non-adjacent interior angles.
2. In any triangle, at most one of the three angles can be obtuse.
Theorem 4: Two triangles are congruent if two angles and a side of one triangle are respectively equal to two angles and the corresponding side of the other.
If in the above triangles, ∠ B = ∠ B1, ∠ C = ∠ C1, and BC = B1C1, then the two triangles are congruent. This is ASA theorem.
Theorem 5: If two sides of a triangle are equal, then the angles opposite to these sides are equal.
This is the definition of an isosceles triangle. A triangle in which two sides are equal is an isosceles triangle. In other words, if in a triangle, two angles are equal, then it is isosceles. In triangle ABC, if AB = AC, then ∠ ABC = ∠ ACB and vice-versa.
In an isosceles triangle, the median to the base bisects the vertical angle and is perpendicular to the base. In this diagram, if we draw the median AD on base BC, we will have: AB = AC, AD = AD, BC = BC. We already have ∠ ABC = ∠ ACB. Therefore, by SAS test, the two triangles will be congruent. As ∠ ADC = ∠ ADB, each will be 90°. Thus, the median AD is perpendicular to BC.
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