Jnb21 Postulate And Theorems

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1 Side-angel-side (sas) Congruence Postulate If two sides (CA and CB) and the included angle ( BCA ) of a triangle are congruent to the corresponding two sides (C'A' and C'B') and the included angle (B'C'A') in another triangle, then the two triangles are congruent.

Example 1: Let ABCD be a parallelogram and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.

Solution to Example 1: •

In a parallelogram, opposite sides are congruent. Hence sides BC and AD are congruent, and also sides AB and CD are congruent.

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In a parallelogram opposite angles are congruent. Hence angles ABC and CDA are congruent.

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Two sides and an included angle of triangle ABC are congruent to two corresponding sides and an included angle in triangle CDA. According to the

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above postulate the two triangles ABC and CDA are congruent

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2.Side-Side-Side (SSS) Congruence Postulate If the three sides (AB, BC and CA) of a triangle are congruent to the corresponding three sides (A'B', B'C' and C'A') in another triangle, then the two triangles are congruent.

Example 2: Let ABCD be a square and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.

Solution to Example 2: •

In a square, all four sides are congruent. Hence sides AB and CD are congruent, and also sides BC and DA are congruent.

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The two triangles also have a common side: AC. Triangles ABC has three sides congruent to the corresponding three sides in triangle CDA. According to the above postulate the two triangles are congruent. The triangles are also right triangles and isosceles.

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3.Angle-Side-Angle (ASA) Congruence Postulate If two angles (ACB, ABC) and the included side (BC) of a triangle are congruent to the corresponding two angles (A'C'B', A'B'C') and included side (B'C') in another triangle, then the two triangles are congruent.

Example 3: ABC is an isosceles triangle. BB' is the angle bisector. Show that triangles ABB' and CBB' are congruent.

Solution to Example 3: •

Since ABC is an isosceles triangle its sides AB and BC are congruent and also its angles BAB' and BCB' are congruent. Since BB' is an angle bisector, angles ABB' and CBB' are congruent. Two angles and an included side in triangles ABB' are congruent to two corresponding angles and one included side in triangle CBB'. According to the above postulate triangles ABB' and CBB' are congruent.

Angle-Angle-Side (AAS) Congruence Theorem

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If two angles (BAC, ACB) and a side opposite one of these two angles (AB) of a triangle are congruent to the corresponding two angles (B'A'C', A'C'B') and side (A'B') in another triangle, then the two triangles are congruent.

Example 4: What can you say about triangles ABC and QPR shown below.

Solution to Example 4: •

In triangle ABC, the third angle ABC may be calculated using the theorem that the sum of all three angles in a triangle is equal to 180 degrees. Hence angle ABC = 180 - (25 + 125) = 30 degrees

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The two triangles have two congruent corresponding angles and one congruent side. angles ABC and QPR are congruent. Also angles BAC and PQR are congruent. Sides BC and PR are congruent.

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Two angles and one side in triangle ABC are congruent to two corresponding angles and one side in triangle PQR. According to the above theorem they are congruent.

Right Triangle Congruence Theorem

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If the hypotenuse (BC) and a leg (BA) of a right triangle are congruent to the corresponding hypotenuse (B'C') and leg (B'A') in another right triangle, then the two triangles are congruent.

Example 5: Show that the two right triangles shown below are congruent.

Solution to Example 5: •

We first use pythagora’s theorem to find the length of side AB in triangle ABC. length of AB = sq [5 2 - 3 2] = 4

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ABC and B'A'C' are congruent. One leg and the hypotenuse in triangle ABC are congruent to a corresponding leg and hypotenuse in the right triangle A'B'C'.

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According to the above theorem, triangle

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Reflexive Property A quantity is congruent (equal) to itself. a = a Symmetric Property If a = b, then b = a. Transitive Property

If a = b and b = c, then a = c.

Postulates: (asummed to be true).

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Properties:

If equal quantities are added to equal quantities, the sums are equal. If equal quantities are subtracted from equal quantities, the Subtraction Postulate differences are equal. If equal quantities are multiplied by equal quantities, the Multiplication Postulate products are equal. (also Doubles of equal quantities are equal.) If equal quantities are divided by equal nonzero quantities, the Division Postulate quotients are equal. (also Halves of equal quantities are equal.) Substitution Postulate A quantity may be substituted for its equal in any expression. If there is a line and a point not on the line, then there exists one Parallel Postulate line through the point parallel to the given line. Corresponding Angles If two parallel lines are cut by a transversal, then the pairs of Postulate corresponding angles are congruent. Corresponding Angles If two lines are cut by a transversal and the corresponding Converse Postulate angles are congruent, the lines are parallel. Side-Side-Side (SSS) If three sides of one triangle are congruent to three sides of Congruence Postulate another triangle, then the triangles are congruent. If two sides and the included angle of one triangle are Side-Angle-Side (SAS) congruent to the corresponding parts of another triangle, the Congruence Postulate triangles are congruent. If two angles and the included side of one triangle are Angle-Side-Angle (ASA) congruent to the corresponding parts of another triangle, the Congruence Postulate triangles are congruent. Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of Postulate another triangle, the triangles are similar. \Addition Postulate

Theorems: (can be proven true)

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Right Angles Congruent Supplements Congruent Complements Vertical Angles Alternate Interior Angles Alternate Exterior Angles Interiorsion Same Side

Alternate interior Angeles Converse Alternate Exterior Angles Converse Interiors on Same Side Converse Triangle Sum Exterior Angle

All right angles are congruent. If two angles are supplementary to the same angle (or to congruent angles) then the two angles are congruent. If two angles are complementary to the same angle (or to congruent angles) then the two angles are congruent. Vertical angles are congruent. If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel. If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel. The sum of the interior angles of a triangle is 180º. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Angle-Angle-Side (AAS) Congruence Base Angle Theorem (Isosceles Triangle) Base Angle Converse (Isosceles Triangle) Hypotenuse-Leg (HL) Congruence (right triangle) Mid-segment Theorem (also called mid-line) Parallelograms

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. If two sides of a triangle are congruent, the angles opposite these sides are congruent. If two angles of a triangle are congruent, the sides opposite these angles are congruent. If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent. The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. * If a quadrilateral is a parallelogram, the About Sides opposite sides are parallel. * If a quadrilateral is a parallelogram, the opposite sides are congruent.

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* If a quadrilateral is a parallelogram, the opposite About Angles angles are congruent. * If a quadrilateral is a parallelogram, the consecutive angles are supplementary. * If a quadrilateral is a parallelogram, the diagonals About bisect each other. Diagonals * If a quadrilateral is a parallelogram, the diagonals form two congruent triangles. * If both pairs of opposite sides of a quadrilateral are parallel, the quadrilateral is a parallelogram. * If both pairs of opposite sides of a About Sides quadrilateral are congruent, the quadrilateral is a Parallelogram Converses

Side Proportionality

parallelogram. * If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a About Angles parallelogram. * If the consecutive angles of a quadrilateral are supplementary, the quadrilateral is a parallelogram. * If the diagonals of a quadrilateral bisect each other, the quadrilateral is a About parallelogram. Diagonals * If the diagonals of a quadrilateral form two congruent triangles, the quadrilateral is a parallelogram. If two triangles are similar, the corresponding sides are in proportion.