Electron microscopes cannot clearly depict the exact shape and structure of atoms and molecules, even though it does show a vague, cloudy image. In my AP chemistry class, I learned that the bond angle of some molecules is 109.5 degrees. How is this bond angle determined so precisely, if the bonds cannot be accurately observed through a microscope?
The positions of atoms respective to each other in a crystal lattice (solid) can be determined by X-ray crystallography. From these positions bond lengths and bond angles can also be calculated accurately.
Probably the most memorable case of solving the geometrical structure of a molecule was Franklin and Gosling's X-ray crystallography of DNA, information later used by Watson and Crick to solve the mystery of DNA's structure.
For many simple (binary) compounds molecular shapes and bond angles can also be determined theoretically (see link).
We use valence-shell electron-pair repulsion (VSEPR) model to predict the geometry of covalently bonded molecules and ions. After the predicted model and angles are determined we perform calculations using the non-relativistic Schrodinger equation based on the vibrational modes of the molecule and compare those to the spectroscopic data for agreement.
Quoting from this tutorial,
The VSEPR model can be explained in the following way. We know that an atom has an outer shell of valence electrons. These valence electrons may be involved in the formation of single, double, or triple bonds, or they may be unshared. Each set of electrons, whether unshared or in a bond, creates a negatively charged region of space. We have already learned that like charges repel each other. The VSEPR model states that the various regions containing electrons or electron clouds around an atom spread out so that each region is as far from the others as possible.
You mention an angle of 109.5 degrees. This angle refers to Structures with Four Regions of High Electron Density around the Central Atom.
The following Lewis structures show three molecules whose central atom is surrounded by four clouds of high electron density:
Quoting again from here,
These molecules are alike in that each central atom is surrounded by four pairs of electrons, but they differ in the number of unshared electron pairs on the central atom. Remember that, although we have drawn them in a plane, the molecules are three-dimensional and atoms may be in front of or behind the plane of the paper. What geometry does the VSEPR theory predict for these molecules?
Let us predict the shape of methane, CH4. The Lewis structure of methane shows a central atom surrounded by four separate regions of high electron density. Each region consists of a pair of electrons bonding the carbon atom to a hydrogen atom. According to the VSEPR model, these regions of high electron density spread out from the central carbon atom in such a way that they are as far from one another as possible.
You can predict the resulting shape using a styrofoam ball or marshmallow and four toothpicks. Poke the toothpicks into the ball, making sure that the free ends of the toothpicks are as far from one another as possible. If you have positioned them correctly, the angle between any two toothpicks will be 109.5°. If you now cover this model with four triangular pieces of paper, you will have built a four-sided figure called a regular tetrahedron. Figure 7.8 shows (a) the Lewis structure for methane, (b) the tetrahedral arrangement of the four regions of high electron density around the central carbon atom, and (c) a space-filling model of methane.
Once you predict the appropriate bond angle from the VSEPR model then based on this model, one can begin to perform calculations of energy associated with different vibrational modes of the molecule using the non-relativistic Schrodinger equation. One then compares those results to those values observed in spectroscopic data that verify that the model is correct.