applying congruent triangles “six steps to success”

Applying Congruent Applying Congruent TrianglesTriangles

““Six Steps To Success”Six Steps To Success”

5-1 Special Segments in 5-1 Special Segments in TrianglesTriangles

Any point on the perpendicular bisector of a segment is equidistant from the endpointsAny point on the perpendicular bisector of a segment is equidistant from the endpoints

So…AP is congruent to BP!So…AP is congruent to BP!

5-1 Special Segments in 5-1 Special Segments in TrianglesTriangles

Stated another way, any point equidistant from the endpoints of a Stated another way, any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.segment lies on the perpendicular bisector of the segment.

If AP is congruent If AP is congruent with BP, then P is with BP, then P is on the perpendicular on the perpendicular bisectorbisector

5-1 Special Segments in 5-1 Special Segments in TrianglesTrianglesAny point on the bisector of an angle is equidistant from the sides of the angle.Any point on the bisector of an angle is equidistant from the sides of the angle.

WE and AB areWE and AB are perpendicularperpendicular

WE is congruentWE is congruent with WFwith WF

5-2 Right Triangles5-2 Right Triangles

LL Theorem – To prove two right triangles LL Theorem – To prove two right triangles congruent when you know the two legs.congruent when you know the two legs.

5-2 Right Triangles5-2 Right Triangles

HA Theorem - To prove two right triangles HA Theorem - To prove two right triangles congruent when you know the hypotenuse congruent when you know the hypotenuse and an acute angle of both triangles.and an acute angle of both triangles.

5-2 Right Triangles5-2 Right TrianglesLA Theorem - To prove two right LA Theorem - To prove two right

triangles congruent when you know the triangles congruent when you know the leg and an acute angle of both triangles.leg and an acute angle of both triangles.

5-2 Right Triangles5-2 Right TrianglesHL Postulate - To prove two right HL Postulate - To prove two right

triangles congruent when you know the triangles congruent when you know the hypotenuse and leg of both triangles.hypotenuse and leg of both triangles.

5-3 Indirect Proof & 5-3 Indirect Proof & InequalitiesInequalities

Steps for writing an Indirect Proof:Steps for writing an Indirect Proof:1.1. Assume that the conclusion is falseAssume that the conclusion is false

2.2. Show that the assumption leads to a Show that the assumption leads to a contradiction of the hypothesiscontradiction of the hypothesis

3.3. Point out that the assumption must be Point out that the assumption must be false, and therefore the conclusion must be false, and therefore the conclusion must be true.true.

Definition of Inequality – a relationship Definition of Inequality – a relationship between two numbers that are not between two numbers that are not equal to each other. (Example: < or >)equal to each other. (Example: < or >)


Exterior Angle Inequality Theorem – Exterior Angle Inequality Theorem – “If an angle is an exterior angle of a “If an angle is an exterior angle of a triangle, then its measure is greater triangle, then its measure is greater than the measure of either remote than the measure of either remote interior angle. interior angle.

Angle 4 > Angle 2Angle 4 > Angle 2 or or Angle 4 > Angle 4 > Angle 3Angle 3


Working Backward – after assuming Working Backward – after assuming that the conclusion is false, you work that the conclusion is false, you work backward from the assumption to backward from the assumption to show that for the given information, show that for the given information, the assumption itself is false. the assumption itself is false.

Example: What is the original number Example: What is the original number if you multiply it by three and then if you multiply it by three and then add nine to get thirty?add nine to get thirty?

5-4 Inequalities For Triangles5-4 Inequalities For Triangles If one side of a triangle is longer than If one side of a triangle is longer than

another side, then the angle opposite another side, then the angle opposite the first side will be greater than the the first side will be greater than the angle opposite the second. angle opposite the second.

If BC > AB, then angle A > angle CIf BC > AB, then angle A > angle C

5-4 Inequalities For Triangles5-4 Inequalities For Triangles If one angle of a triangle is longer than If one angle of a triangle is longer than

another angle, then the side opposite another angle, then the side opposite the first angle will be greater than the the first angle will be greater than the side opposite the second angle. side opposite the second angle.

If angle A > angle C, then BC > ABIf angle A > angle C, then BC > AB

5-4 Inequalities For Triangles5-4 Inequalities For TrianglesThe perpendicular segment from a The perpendicular segment from a

point to a line is the shortest point to a line is the shortest segment from the point to the line. segment from the point to the line.

ExampleExample::

What is the shortestWhat is the shortest

distance betweendistance between

ST and point V?ST and point V?

5-4 Triangle Inequality5-4 Triangle InequalityTriangle Inequality Theorem – the Triangle Inequality Theorem – the

sum of the lengths of any two sides sum of the lengths of any two sides of a triangle is greater than the of a triangle is greater than the length of the third side.length of the third side.

ExampleExample::

If sides of a figure are 15, 32, and 16 If sides of a figure are 15, 32, and 16 could the figure be a triangle?could the figure be a triangle?

AnswerAnswer: NO! (15 + 16 is not > 32): NO! (15 + 16 is not > 32)

5-4 Triangle Inequality5-4 Triangle InequalityExampleExample::


AnswerAnswer: NO! (3 + 7 is not > 12): NO! (3 + 7 is not > 12)

ExampleExample::


AnswerAnswer: Yes! (17 + 22 > 34): Yes! (17 + 22 > 34)

Chapter 5 ProofsChapter 5 Proofs

Chapter 5 ProofsChapter 5 ProofsMore to come soon!!!More to come soon!!!

applying congruent triangles “six steps to success”

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