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Stages of solving geometry problems
Textbook: pp. 144–145
GoalKnow the four stages of solving a geometry problem and the types of problems.
New words
given conditions · masala sharticonclusion · xulosadiagram · chizmachecking · tekshirish
Explanation
It pays to solve a geometry problem in order. Stage 1: understand the problem. Write separately what is given and what is to be found, proved or constructed, draw a clear diagram and mark the data on it. Stage 2: plan. Choose the theorem or method and, if needed, draw an auxiliary line. Stage 3: solve, that is, work according to the plan. Stage 4: check. See whether the result fits the conditions and correct any mistake. Problems are of three kinds: calculation, proof and construction.
Worked examples
Calculation. In triangle ABC, ∠A = 48° and ∠B is twice ∠A; find ∠C. Understand: given ∠A = 48°, ∠B = 2·∠A; to find ∠C. Plan: the angles of a triangle sum to 180°. Solve: ∠B = 96°, ∠C = 180° − 48° − 96° = 36°. Check: 48° + 96° + 36° = 180°.
Proof. In isosceles triangle ABC with AB = AC, D is the midpoint of BC. Prove ∠BAD = ∠CAD. Plan: compare △ABD and △ACD by SSS. Solve: AB = AC (given), BD = DC (D is the midpoint), AD is common. So △ABD = △ACD and corresponding angles are equal: ∠BAD = ∠CAD. Check: every step rests on a condition or a theorem.
Class activity
“Stage cards”: write the four stage names on four slips, shuffle them and put them in order. Then solve one simple problem following the cards.
Practice
1
How many stages are there in solving a geometry problem?
4
2
Into how many kinds are geometry problems divided?
3
3
A triangle has angles 50°, 60° and the found third one 70°. Find the sum for checking.
180
4
Why should the diagram be large and clear with the data marked on it?
A diagram shows the conditions and helps to see which theorem is needed.