MO Z7–I–6 2021

In the triangle ABC, point D lies on the AC side and point E on the BC side. The sizes of the angles ABD, BAE, CAE and CBD are 30°, 60°, 20° and 30°, respectively. Find the size of the AED angle.

Correct answer:

AED =  20 °

Step-by-step explanation:

$$\begin{gathered} \mathrm{\angle }ABD=30^{\circ } \\ \mathrm{\angle }BAE=60^{\circ } \\ \mathrm{\angle }CAE=20^{\circ } \\ \mathrm{\angle }CBD=30^{\circ } \\ \alpha =\mathrm{\angle }CAE+\mathrm{\angle }BAE=20+60=80^{\circ } \\ \beta =\mathrm{\angle }ABD+\mathrm{\angle }CBD=30+30=60^{\circ } \\ \mathrm{\angle }ADB=180-\alpha -\mathrm{\angle }ABD=180-80-30=70^{\circ } \\ \gamma =180-\alpha -\beta =180-80-60=40^{\circ } \\ \mathrm{\angle }AEB=180-\mathrm{\angle }BAE-\beta =180-60-60=60^{\circ } \\ \mathrm{\angle }ASB=180-\mathrm{\angle }BAE-\mathrm{\angle }ABD=180-60-30=90^{\circ } \\ AED+BDE=90 \\ \mathrm{\angle }ADB=180-\mathrm{\angle }CAE-(AED+BDE) \\ \mathrm{\angle }ADB=180-\mathrm{\angle }CAE-90=180-20-90=70^{\circ } \\ \mathrm{\angle }AEC=180-\gamma -\mathrm{\angle }CAE=180-40-20=120^{\circ } \\ \mathrm{\angle }CDB=180-\gamma -\mathrm{\angle }CBD=180-40-30=110^{\circ } \\ \mathrm{\mid }AS\mathrm{\mid }=\mathrm{\mid }ES\mathrm{\mid } \\ \mathrm{\Delta }\mathrm{\angle }AEB:60^{\circ }-60^{\circ }-60^{\circ } \\ \mathrm{\Delta }ASD\dots DSE \\ AED=\mathrm{\angle }CAE=20=20^{\circ } \end{gathered}$$

Try another example

Related math problems and questions:

• Internal angles
  The ABCD is an isosceles trapezoid, which holds: |AB| = 2 |BC| = 2 |CD| = 2 |DA|: On its side BC is a K point such that |BK| = 2 |KC|, on its side CD is the point L such that |CL| = 2 |LD|, and on its side DA the point M is such that | DM | = 2 |MA|. Dete
• Equilateral triangle ABC
  In the equilateral triangle ABC, K is the center of the AB side, the L point lies on one-third of the BC side near the point C, and the point M lies in the one-third of the side of the AC side closer to the point A. Find what part of the ABC triangle cont
• Similarity coefficient
  The triangles ABC and A "B" C "are similar to the similarity coefficient 2. The sizes of the angles of the triangle ABC are α = 35° and β = 48°. Find the magnitudes of all angles of triangle A "B" C ".
• Triangles
  Find out whether given sizes of the angles can be interior angles of a triangle: a) 23°10',84°30',72°20' b) 90°,41°33',48°37' c) 14°51',90°,75°49' d) 58°58',59°59',60°3'
• Two angles
  The triangles ABC and A'B'C 'are similar. In the ABC triangle, the two angles are 25° and 65°. Explain why in the triangle A'B'C 'is the sum of two angles of 90 degrees.
• The triangles
  The triangles ABC and A'B'C 'are similar with a similarity coefficient of 2. The angles of the triangle ABC are alpha = 35°, beta = 48°. Determine the magnitudes of all angles of triangle A'B'C '.
• MO - triangles
  On the AB and AC sides of the triangle ABC lies successive points E and F, on segment EF lie point D. The EF and BC lines are parallel and is true this ratio FD:DE = AE:EB = 2:1. The area of ABC triangle is 27 hectares and line segments EF, AD, and DB seg
• Area and two angles
  Calculate the size of all sides and internal angles of a triangle ABC, if it is given by area S = 501.9; and two internal angles α = 15°28' and β = 45°.
• Outer angles
  The outer angle of the triangle ABC at the A vertex is 71°40 ' outer angle at the vertex B is 136°50'. What size has the inner triangle angle at the vertex C?
• Inner angles
  The magnitude of the internal angle at the main vertex C of the isosceles triangle ABC is 72°. The line p, parallel to the base of this triangle, divides the triangle into a trapezoid and a smaller triangle. How big are the inner angles of the trapezoid?
• A kite
  ABCD is a kite. Angle OBC = 20° and angle OCD = 35°. O is the intersection of diagonals. Find angle ABC, angle ADC and angle BAD.
• Similarity
  Are two right triangles similar to each other if the first one has an acute angle 70°, and the second one has an acute angle 20°?
• Supplementary angles
  One of the supplementary angles is larger by 33° than the second one. Calculate the angles size.
• Tropics and polar zones
  What percentage of the Earth's surface lies in the tropical, temperate, and polar zone? Individual zones are bordered by tropics 23°27' and polar circles 66°33'.
• Isosceles - isosceles
  It is given a triangle ABC with sides /AB/ = 3 cm /BC/ = 10 cm, and the angle ABC = 120°. Draw all points X such that true that BCX triangle is an isosceles and triangle ABX is isosceles with the base AB.
• Right triangle trigonometrics
  Calculate the size of the remaining sides and angles of a right triangle ABC if it is given: b = 10 cm; c = 20 cm; angle alpha = 60° and the angle beta = 30° (use the Pythagorean theorem and functions sine, cosine, tangent, cotangent)
• Acute triangle
  In the acute triangle KLM, V is the intersection of its heights and X is the heel of height to the side KL. The axis of the angle XVL is parallel to the side LM and the angle MKL is 70°. What size are the KLM and KML angles?