كتاب الطالب الرياضيات Reveal الصف التاسع منهج انجليزي الفصل الأول
Explore Applying Indirect Reasoning
Online Activity Use the video to complete the Explore
INQUIRY How can you use a contradiction to prove a conclusion
Learn Indirect Proof A direct proof is one that starts with a true hypothesis, and the conclusion is proved to be true. Indirect reasoning eliminates all possible conclusions but one, so the one remaining conclusion must be true. In an indirect proof, or proof by contradiction, one assumes that the statement to be proved is false and then uses logical reasoning to deduce that a statement contradicts a postulate, theorem, or one of the assumptions. Once a contradiction is obtained, one concludes that the statement assumed false must in fact be true
Key Concept - How to Write an Indirect Proof
Step 1 ldentify the conclusion that you are asked to prove. Make the assumption that this conclusion is false by assuming that the negation is true
Step 2 Use logical reasoning to show that this assumption leads to a contradiction of the hypothesis or some other fact such as a definition, postulate, theorem, or corollary
Step 3 State that because the assumption leads to a contradiction, the original conclusion, what you were asked to prove, must be true
In indirect proofs, you should assume that the conclusion you are trying to prove is false. If, in the proof, you prove that the hypothesis is then false, this is a proof by contrapositive. If, in the proof, you assume that the hypothesis is true and prove that some other known fact is false, this is a proof by contradiction
Example 1 Write an Indirect Algebraic Proof Write an indirect proof to show that
Module Summary
Lessons 1-1 and 1-2
Perpendicular Bisectors and Angle Bisectors
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
The perpendicular bisectors of a triangle intersect at the circumcenter that is equidistant from the vertices of the triangle
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle
The angle bisectors of a triangle intersect at the incenter, which is equidistant from the sides of the triangle
Lesson 1-3
Medians and Altitudes
A median of a triangle is a line segment with endpoints that are a vertex of the triangle and the midpoint of the side opposite the vertex. The medians of a triangle intersect at the centroid, which is two- thirds of the distance from each vertex to the midpoint of the opposite side
An altitude of a triangle is a segment from a vertex of the triangle to the line that contains the opposite side and is perpendicular to that side
The altitudes of a triangle intersect at a point called the orthocenter
Lessons 1-4, 1-6, and 1-7
Inequalities in Triangles
. If one side of a triangle is longer than another side, then the angle that is opposite the longer side has a greater measure than the angle that is opposite the shorter side
The sum of the lengths of any two sides of a triangle. must be greater than the length of the third side
If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle
Lesson 1- 5
Indirect Proof
To write an indirect proof Identify the conclusion you are asked to prove. Make the assumption that this conclusion is false by assuming that the opposite is true
Use logical reasoning to show that this assumption leads to a contradiction of the hypothesis or some other fact such as a definition, postulate, theorem, or corollary. State that because the assumption leads to a contradiction, the original conclusion, what you were asked to prove, must be true
. ARCHAEOLOGY Archaeologists unearthed parts of two adjacent walls of an ancient castle. Before it was unearthed, they knew from ancient texts that the castle was shaped like a regular polygon, but nobody knew how many sides it had. Some said 6, others 8, and some even said 100. From the information in the figure, how many sides did the castle really have
. DESIGN Ronella is designing boxes she will use to ship her jewelry. She wants to
shape the box like a regular polygon. For the boxes to pack tightly, she decides to use a regular polygon in which the measure of its interior angles is half the measure of its exterior angles. What regular polygon should she use
. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems The basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to the triclinic system. Each of the six faces of turquoise is in the shape of a quadrilateral. Find the sum of the measures of the interior angles of one such face
. STRUCTURE If three of the interior angles of a convex hexagon each measure
140', a fourth angle measures 84', and the measure of the fifth angle is 3 times the measure of the sixth angle, find the measure of the sixth angle
. FIND THE ERROR Marshawn says that the sum of the exterior angles of a decagon is greater than that of a heptagon because a decagon has more sides. Liang says that the sum of the exterior angles for both polygons is the same. Who is correct ? Explain your reasoning
. WRITE Explain how triangles are related to the Polygon Interior Angles Sum Theorem
. CREATE Sketch a polygon and find the sum of its interior angles. How many sides does a polygon with twice this interior angles sum have ? Justify your answer
. PERSEVERE Find the values of a, b, and cif QRSTVX is a regular hexagon. Justify your answer
. ANALYZE If two sides of a regular hexagon are extended to meet at a point in the exterior of the polygon, will the triangle formed sometimes, always, or never be equilateral ? Justify your argument