Math Problem Statement

Find the measure of the three missing angles in the rhombus where one angle is 78°.

Solution

To solve for the missing angles xx^\circ, yy^\circ, and zz^\circ in the rhombus, we can use the following properties:

  1. All angles in a quadrilateral add up to 360°.
  2. Opposite angles in a rhombus are equal.
  3. Adjacent angles in a rhombus are supplementary (add up to 180°).

From the image:

  • One angle is given as 7878^\circ at the bottom left.
  • Let the opposite angle at the top right be xx^\circ, so x=78x^\circ = 78^\circ (since opposite angles are equal).
  • The adjacent angles yy^\circ and zz^\circ must each satisfy 78+z=18078^\circ + z^\circ = 180^\circ and 78+y=18078^\circ + y^\circ = 180^\circ because they are supplementary.

Thus:

  • z=18078=102z^\circ = 180^\circ - 78^\circ = 102^\circ
  • y=18078=102y^\circ = 180^\circ - 78^\circ = 102^\circ

Therefore, the three missing angles are:

  • x=78x^\circ = 78^\circ
  • y=102y^\circ = 102^\circ
  • z=102z^\circ = 102^\circ

Would you like more details on any of the steps? Here are five related questions to expand your understanding:

  1. What other properties of rhombuses can help in solving geometry problems?
  2. How would this solution change if the given angle was different, like 90° or 120°?
  3. How do the properties of parallelograms apply to rhombuses?
  4. What is the relationship between the diagonals of a rhombus and its angles?
  5. How do you derive the total sum of interior angles for any polygon?

Tip: Always remember that opposite angles of a rhombus are equal, and adjacent angles are supplementary. This simplifies the angle calculations significantly!

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Math Problem Analysis

Mathematical Concepts

Geometry
Properties of Rhombus
Quadrilaterals

Formulas

Sum of interior angles of a quadrilateral = 360°

Theorems

Opposite angles of a rhombus are equal
Adjacent angles in a rhombus are supplementary

Suitable Grade Level

Grades 7-9