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Chapter 1: Problem 29

Problem \(29(A+B=C)\) The 3 by 1 rectangle \(A D E H\) consists of three adjacent unit squares: \(A B G H, B C F G, C D E F\) left to right, with \(A\) in the top left corner. Prove that $$ \angle D A E+\angle D B E=\angle D C E $$

Short Answer

Expert verified
The sum of \(\angle DAE\) and \(\angle DBE\) equals \(\angle DCE\) because of the supplementary properties of angles on a straight line.

Step by step solution

01

Understanding the Problem Setup

Visualize the rectangle as consisting of three unit squares placed side by side, forming a single rectangular block. The points are as follows: \(A, B, C, D\) are the top corners from left to right, and \(H, G, F, E\) are the bottom corners from left to right. The square \(ABGH\) is on the left, \(BCFG\) in the middle, and \(CDEF\) on the right.
02

Identify the Points and Lines

Identify that \(\triangle DAE\), \(\triangle DBE\), and \(\triangle DCE\) are the triangles formed within the problem. Notice that they all share a common vertex \(E\) and a common point \(D\).
03

Establish Relationships Between Angles

These triangles share side \(DE\). Since \(AE = AH = 1\), \(BE = BF = 1\), and \(CE = CD = 1\), each side represents one unit. Use properties of supplemental angles on a straight line (i.e., angles on a straight line sum to 180掳).
04

Apply Angle Sum Property in Triangles

In triangle \(\triangle DCE\), \(\angle DCE + \angle DAE + \angle DBE = 180^\circ\). Since \(\angle DAE + \angle DBE = \angle DCE\), the alternate interior angles formed by extending the base DE must also obey this relationship.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangles
Triangles are three-sided polygons that are fundamental in geometry. They offer a way to understand shapes and solve many geometric problems. Each triangle consists of three edges and three vertices.
  • Vertices: The corners where two sides meet. For example, in our exercise, the triangles have vertices like \(D, A, E\).
  • Edges: The line segments connecting the vertices, such as \(DE\) in triangle \(\triangle DCE\).
Triangles can be classified by their sides or their angles:
  • Equilateral: All sides and angles are equal.
  • Isosceles: Two sides and their opposite angles are equal.
  • Scalene: No sides or angles are the same.
In our problem, understanding how triangles \(\triangle DAE\), \(\triangle DBE\), and \(\triangle DCE\) interact is crucial for proving the angles' relationship.
Angle Sum Property
The angle sum property of triangles states that the sum of the interior angles in a triangle is always 180 degrees. This concept is a key player in solving many triangle-related problems, including the one in our example.
To see this in action:
  • In triangle \(\triangle DCE\), the interior angles are \(\angle DCE\), \(\angle DAE\), and \(\angle DBE\).
  • According to the angle sum property, \(\angle DCE + \angle DAE + \angle DBE = 180^\circ\).
This equation is central to the proof, as it lays down the mathematical path to show how the angles relate to each other. By understanding this property, we can split and rearrange angles to show the desired equality.
Supplemental Angles
Supplemental angles are two angles whose measures add up to 180 degrees. These often appear in problems involving a straight line or when angles form a linear pair.
  • A good example is when two angles lie on a straight line, such as \(\angle DAE\) and an adjacent angle forming a linear pair with \(\angle DCE\).
  • If their sum is 180 degrees, they are supplemental.
In our exercise, recognizing supplemental angles helps in understanding how \(\angle DAE\), \(\angle DBE\), and \(\angle DCE\) connect through shared boundaries or lines.
This understanding is used to manipulate the angle expressions in the solutions to reach our prove statement correctly.
Unit Squares
A unit square is a square with a side length of one unit, which serves as a fundamental building block in geometry, especially in grid-like setups. Each side being equal simplifies calculations.
  • In the exercise, the rectangle \(ADEH\) is composed of three unit squares: \(ABGH\), \(BCFG\), and \(CDEF\).
  • The key property here is that each unit square鈥檚 side lengths are exactly one unit.
This consistent measurement allows the problem solver to use properties of equal sides to establish relationships between various lengths.
Moreover, having uniform squares aids in visualizing the spatial arrangement, ensuring clarity when identifying important points like \(A, B, C, D, E, F, G,\) and \(H\).
This visualization is instrumental in finding and assembling the triangles needed for proving the problem's angle relationships.

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Most popular questions from this chapter

Let \(A B C\) be a triangle. We use the standard labelling convention, whereby the side \(B C\) opposite \(A\) has length \(a\), the side \(C A\) opposite \(B\) has length \(b,\) and the side \(A B\) opposite \(C\) has length \(c\). Prove that, if \(c^{2}=a^{2}+b^{2},\) then \(\angle B C A\) is a right angle.

(a) The operation of "squaring" is a function: it takes a single real number \(x\) as input, and delivers a definite real number \(x^{2}\) as output. \- Every positive number arises as an output ("is the square of something" ). \(-\) Since \(x^{2}=(-x)^{2},\) each output (other than 0 ) arises from at least two different inputs. \- If \(a^{2}=b^{2},\) then \(0=a^{2}-b^{2}=(a-b)(a+b)\), so either \(a=b\), or \(a=-b\). Hence no two positive inputs have the same square, so each output (other than 0 ) arises from exactly two inputs (one positive and one negative). \- Hence each positive output \(y\) corresponds to just one positive input, called \(\sqrt{y}\). Find: (i) \(\sqrt{49}\) (ii) \(\sqrt{144}\) (iii) \(\sqrt{441}\) (iv) \(\sqrt{169}\) (v) \(\sqrt{196}\) (vi) \(\sqrt{961}\) (vii) \(\sqrt{96100}\) (b) Let \(a>0\) and \(b>0\). Then \(\sqrt{a b}>0\), and \(\sqrt{a} \times \sqrt{b}>0\), so both expressions are positive. Moreover, they have the same square, since $$ (\sqrt{a b})^{2}=a b=(\sqrt{a})^{2} \cdot(\sqrt{b})^{2}=(\sqrt{a} \times \sqrt{b})^{2} $$ \(\therefore \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}\) Use this fact to simplify the following: (i) \(\sqrt{8}\) (ii) \(\sqrt{12}\) (iii) \(\sqrt{50}\) (iv) \(\sqrt{147}\) (v) \(\sqrt{288}\) (vi) \(\sqrt{882}\) (c) [This part requires some written calculation.] Exact expressions involving square roots occur in many parts of elementary mathematics. We focus here on just one example - namely the regular pentagon. Suppose that a regular pentagon \(A B C D E\) has sides of length \(1 .\) (i) Prove that the diagonal \(A C\) is parallel to the side \(E D\). (ii) If \(A C\) and \(B D\) meet at \(X,\) explain why \(A X D E\) is a rhombus. (iii) Prove that triangles \(A D X\) and \(C B X\) are similar. (iv) If \(A C\) has length \(x\), set up an equation and find the exact value of \(x\). (v) Find the exact length of \(B X\). (vi) Prove that triangles \(A B D\) and \(B X A\) are similar. (vii) Find the exact values of \(\cos 36^{\circ}, \cos 72^{\circ}\). (viii) Find the exact values of \(\sin 36^{\circ}, \sin 72^{\circ}\).

The 4 by 4 "multiplication table" below is completely familiar. \(\begin{array}{rrrr}1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\ 3 & 6 & 9 & 12 \\ 4 & 8 & 12 & 16\end{array}\) What is the total of all the numbers in the 4 by 4 square? How should one write this answer in a way that makes the total obvious?

[This problem requires a mixture of serious thought and written proof.] (a) I choose six integers between 10 and 19 (inclusive). (i) Prove that some pair of integers among my chosen six must be relatively prime. (ii) Is it also true that some pair must have a common factor? (b) I choose six integers in the nineties (from \(90-99\) inclusive). (i) Prove that some pair among my chosen integers must be relatively prime. (ii) Is it also true that some pair must have a common factor? (c) I choose \(n+1\) integers from a run of \(2 n\) consecutive integers. (i) Prove that some pair among the chosen integers must be relatively prime. (ii) Is it also true that some pair must have a common factor?

(a)(i) Expand \((a+b)^{2}\) and \((a+b)^{3}\). (ii) Without doing any more work, write out the expanded forms of \((a-b)^{2}\) and \((a-b)^{3}\). (b) Factorise (i) \(x^{2}+2 x+1\) (ii) \(x^{4}-2 x^{2}+1\) (iii) \(x^{6}-3 x^{4}+3 x^{2}-1\). (c)(i) Expand \((a-b)(a+b)\). (ii) Use (c)(i) and (a)(i) to write down (with no extra work) the expanded form of $$ (a-b-c)(a+b+c) $$ and of $$ (a-b+c)(a+b-c) $$ (d) Factorise \(3 x^{2}+2 x-1\).
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