Review questions for Chapter 2. Review questions for Chapter II. Additional tasks for Chapter II
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Geometry 7th grade. Review questions for Chapter II. Help please
Answers:
A triangle is a figure consisting of three points that do not lie on the same straight line and three segments connecting these points. The perimeter of a triangle is the sum of the lengths of the three sides of the triangle. (Figure in attachment) No. 2 Equal triangles are those triangles in which the corresponding elements (sides and angles) are equal ) No. 3 A theorem is a statement whose validity is established by reasoning, and the reasoning itself is called proof of the theorem. No. 4 The first sign of equality of triangles If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are equal. Proof p. 30. No. 5 The segment AN is called a perpendicular drawn from point A to line a if the lines AN and a are perpendicular. Drawing on page 32 (Fig. 55) (Fig. 55) No. 6 Theorem From a point not lying on a line, you can draw a perpendicular to this line, and only one. (proof page 32) No. 7 The segment connecting the vertex of the triangle with the midpoint of the opposite side is called the median of the triangle. In total, the triangle has 3 medians. No. 8 the segment, bisectors of the angle of the triangle connecting the vertex of the triangle with the point of the opposite side is called the bisector of the triangle. A triangle has three bisectors. No. 9 Perpendicular, drawn from the vertex of a triangle to a line containing the opposite side is called the altitude of the triangle. Any triangle has three altitudes. No. 10 A triangle is called isosceles if its two sides are equal. Equal sides are called lateral sides, and the third side is called the base. No. 11 A triangle whose all sides are equal is called equilateral. No. 12 Proof on page 35 No. 13 Theorem In an isosceles triangle, the bisector drawn to the base is the median and altitude (proof pp. 35-36) No. 14 If the side and two adjacent angles of one triangle are respectively equal to the side and two adjacent angles of another triangle, then such triangles are congruent. (proof on pages 38-39) No. 15 If three sides of a triangle are respectively equal to three sides of another triangle, then such triangles are congruent . (proof 39-40 pages) No. 16 Definition - sentence, which explains the meaning of a particular expression or name. Circle is a geometric figure consisting of all points of the plane located at a given distance from a given point. Center is a given point. radius is a segment connecting the center with any point on the circle chord is a segment connecting two points on the circle diameter- chord passing through the center. Answers to questions in Chapter III No. 1 Two lines are called parallel if they do not intersect. Two segments are called parallel if they lie on parallel lines. No. 2 Line c is called a secant with respect to lines a and b if it intersects them in two points. angles are formed: opposite, one-sided and corresponding. No. 7 axiom - initial positions examples: through any two points there is a straight line and, moreover, only one on any ray from its beginning, you can plot a segment equal to the given one and, moreover, only one. No. 9 through a point not lying on given a straight line there is only one straight line parallel to a given one No. 10 Corollary - statements that are derived directly from axioms or theorems No. 12 An inverse theorem is a theorem in which the condition is the conclusion of this theorem, and the conclusion is the condition of this theorem. Example: if two parallel lines are intersected by a transversal, then the opposite angles are equal. Answers to questions for review for Chapter IV No. 1 The sum of the angles of a triangle is 180 degrees No. 2 An external angle is an angle adjacent to some angle of this triangle. No. 4 an acute triangle is a triangle if all its angles are acute; an obtuse triangle is a triangle if one of its angles is obtuse; No. 5 a right triangle is a triangle in which one of its angles is right. Side lying against right angle is called the hypotenuse, the other two are called legs. No. 9 The triangle inequality follows from the corollary: For any three points A, B, C not lying on the same line, the inequalities AB are valid
1. Explain what figure is called a triangle. Draw a triangle and show its sides, vertices and angles. What is the perimeter of a triangle?
2. Which triangles are called equal?
3. What is a theorem and proof of the theorem?
4. Formulate and prove a theorem expressing the first criterion for the equality of triangles.
5. Explain what segment is called a perpendicular drawn from a given point to a given line.
6. Formulate and prove a theorem about the perpendicular drawn from a given point to a given line.
7. Which segment is called the median of a triangle? How many medians does a triangle have?
8. Which segment is called the bisector of a triangle? How many bisectors does a triangle have?
9. Which segment is called the altitude of the triangle? How many heights does a triangle have?
10. Which triangle is called isosceles? What are its sides called?
11. Which triangle is called equilateral?
12. Prove that the angles at the base of an isosceles triangle are equal.
13. State and prove the theorem on the bisector of an isosceles triangle.
14. Formulate and prove a theorem expressing the second criterion for the equality of triangles.
15. Formulate and prove a theorem expressing the third criterion for the equality of triangles.
16. What is the definition? Define a circle. What are the center, radius, chord and diameter of a circle?
17. Explain how to plot a segment equal to the given one on a given ray from its beginning.
18. Explain how to plot an angle equal to a given one from a given ray.
19. Explain how to construct the bisector of a given angle.
20. Explain how to construct a line passing through this point lying on a given line and perpendicular to this line.
21. Explain how to construct the midpoint of this segment.
Additional tasks for Chapter II
156. The perimeter of triangle ABC is 15 cm. Side BC is greater than side AB by 2 cm, and side AB is less than side AC by 1 cm. Find the sides of the triangle.
157. In an isosceles triangle, the base is 2 cm greater than the side side, but less than the sum of the sides by 3 cm. Find the sides of the triangle.
158. The base of an isosceles triangle is 8 cm. The median drawn to the lateral side divides the triangle into two triangles so that the perimeter of one triangle is 2 cm greater than the perimeter of the other. Find the side of this triangle.
159. Prove that two isosceles triangles are equal if side and the angle opposite to the base of one triangle are respectively equal to the lateral side and the angle opposite to the base of the other triangle.
160. Line a passes through the middle of segment AB and is perpendicular to it. Prove that: a) each point on line a is equidistant from points A and B; b) every point equidistant from points A and B lies on line a.
161. In triangles ABC and A 1 B 1 C 1, the medians AM and A 1 M 1 are equal, BC = B 1 C 1 and ∠AMB = ∠A 1 M 1 B 1. Prove that Δ ABC = Δ A 1 B 1 C 1.
162. In Figure 92, triangle ADE is isosceles, DE is the base. Prove that: a) if BD-CE, then ∠CAD = ∠BAE and AB = AC; b) if ∠CAD = ∠BAE, then BD = CE and AB = AC.
Rice. 92
163. Prove that the midpoints of the sides of an isosceles triangle are the vertices of another isosceles triangle.
164. On the sides equilateral triangle ABC contains equal segments AD, BE and CF, as shown in Figure 93. Points D, E, F are connected by segments. Prove that triangle DEF is equilateral.
Rice. 93
165. Segments AB and CD intersect at their common center O. Points K and K 1 are marked on segments AC and BD so that AK = BK 1. Prove that: a) OK = OK 1 ; b) point O lies on the line KK 1.
166. Segments AB and CD intersect at their common midpoint O. Points M and N are the midpoints of segments AC and BD. Prove that point O is the midpoint of the segment MN.
167. The sides of the equilateral triangle ABC are extended, as shown in Figure 94, into equal segments AD, CE, BF. Prove that triangle DEF is equilateral.
Rice. 94
168. In triangle ABC, ∠A = 38°, ∠B = 110°, ∠C = 32°. On side AC, points D and E are marked so that point D lies on the segment AE, BD = DA, BE = EC. Find the angle DBE.
169. In Figure 95 OC = OD, OB = OE. Prove that AB = EF. Explain the method for measuring the width of the lake (segment AB in Figure 95), based on this problem.
Rice. 95
170. Prove that triangles ABC and A 1 B 1 C 1 are equal if AB = A 1 B 1, ∠A = ∠A 1, AD = A 1 D 1, where AD and A 1 D 1 are the bisectors of the triangles.
171. In triangles ABC and ADC, sides BC and AD are equal and intersect at point O, ∠OAC = ∠OCA. Prove that triangles ABO and C DO are congruent.
172. In Figure 96, AC = AD, AB ⊥ CD. Prove that BC = BD and ∠ACB = ∠ADB.
Rice. 96
173.* Prove that an angle adjacent to an angle of a triangle is greater than each of the other two angles of the triangle.
174.* Prove that Δ ABC = Δ A 1 B 1 C 1 if ∠A = ∠A 1 , ∠B = ∠B 1 , BC = B 1 C 1 .
175.* On the sides of the angle XOY points A, B, C and D are marked so that OA = OB, AC = BD (Fig. 97). Lines AD and BC intersect at point E. Prove that ray OE is the bisector of angle XOY. Describe a method for constructing an angle bisector based on this fact.
Rice. 97
176.* Prove that triangles ABC and A 1 B 1 C 1 are equal if AB = A 1 B 1, AC = A 1 C 1, AM = A 1 M 1, where AM and A 1 M 1 are the medians of the triangles.
177.* Given two triangles: ABC and A 1 B 1 C 1. It is known that AB = A 1 B 1, AC = A 1 C 1, ∠A = ∠A 1. On sides AC and BC of triangle ABC, points K and L are taken, respectively, and on sides A 1 C 1 and B 1 C 1 of triangle A 1 B 1 C 1 - points K 1 and L 1 so that AK = A 1 K 1, LC = L 1 C 1 . Prove that: a) KL = K 1 L 1 ; b) AL = A 1 L 1 .
178.* Given three points A, B, C, lying on one line, and point D, not lying on this line. Prove that at least two of the three segments AD, BD and CD are not equal to each other.
179.* On the lateral sides AB and AC of the isosceles triangle ABC, points P and Q are marked so that ∠PXB = ∠QXC, where X is the midpoint of the base BC. Prove that BQ = CP.
180. Construct a circle of given radius passing through a given point with its center on a given line.
181. Construct a circle of given radius passing through two given points.
182. Given a straight line a, points A, B and a segment PQ. Construct triangle ABC so that vertex C lies on line a and AC = PQ.
183. Given a circle, points A, B and segment PQ. Construct triangle ABC so that vertex C lies on the given circle and AC = PQ.
184. On side BC of triangle ABC, construct a point equidistant from vertices A and C.
185. Using a compass and ruler, divide this segment into four equal parts.
Answers to additional problems for Chapter II
156. AB = 4 cm, AC = 5 cm, BC = 6 cm.
157. 7 cm, 5 cm and 5 cm.
158. 10 cm or 6 cm.
160. b) Instruction. Let M be a point equidistant from points A and B and not lying on line AB. Use the statement: the median of an isosceles triangle drawn to the base is the height.
165. b) Instruction. First prove that ∠AOK = ∠BOK 1 .
166. Instruction. Use problem 165.
167. Instruction. First prove the equality of triangles DBF, FCE and EAD.
169. Instruction. Prove that Δ ABO = Δ FEO.
170. Instruction. First prove the equality of triangles ABD and A 1 B 1 D 1 .
171. Instruction. First prove the equality of triangles ABC and ADC.
172. Instruction. First prove the equality of triangles ABC and ABD.
173. Instruction. Let angle BAD be adjacent to angle A of triangle ABC. To prove the inequality ∠BAD > ∠B, mark the middle O of side AB and, on the continuation of the segment CO, plot the segment OE equal to CO. Then prove that angle BAE is equal to angle B of triangle ABC and use the inequality ∠BAD > ∠BAE.
174. Instruction. Place triangle ABC on triangle A 1 B 1 C 1, so that side BC aligns with side B 1 C 1, and side BA overlaps beam BA 1. To prove that point A coincides with point A 1, use problem 173.
175. Instruction. First prove that Δ AOD = Δ BOC, and then that Δ EBD = Δ EAC.
176. Instruction. Consider triangles ABD and A 1 B 1 D 1, where points D and D 1 are such that M and M 1 are the midpoints of segments AD and A 1 D 1.
178. Instruction. Let point B lie on segment AC. Assume that AD = BD = CD. Using the property of angles at the base of an isosceles triangle, first prove that ∠ABD = ∠CBD= 90°.
179. Instruction. First prove that BP = CQ.
184. Instruction. Use problem 160.
